Mendelian Genetics, Punnett Squares and Complete Dominance

1Overview

Why do children resemble their parents, yet differ from them and from each other? Before DNA was known, an Austrian monk named Gregor Mendel answered much of that question with a garden, a lot of patience, and careful counting. His results still form the basis of the way heredity is taught. This lesson covers Mendelian genetics, Punnett squares and complete dominance, the simplest pattern of inheritance, and applies them to human traits.

2Mendel’s experiments with peas

Johann Gregor Mendel (1822–1884) studied inheritance in garden peas, Pisum sativum. Peas are an excellent choice: they have many easily seen traits with two clear forms (violet or white flowers, round or wrinkled seeds), they grow quickly, and they normally self-fertilize, so they produce true-breeding lines in which every generation shows the same trait. He also could cross-pollinate them by hand to control which plants mated. He presented results from nearly 30,000 plants in 1865, and his work was rediscovered in 1900.

Mendel labelled generations: P for the parent plants, F₁ for their offspring and F₂ for the offspring produced when F₁ plants mated with each other.

The pattern was the same for every trait he tested. When true-breeding violet and white plants were crossed, every F₁ plant was violet. The white trait vanished. But when the F₁ plants were crossed, white flowers returned in the F₂ at about one white for every three violet plants (705 violet and 224 white). The same near-3:1 ratio appeared for other traits:

These results contradicted the popular idea of “blending inheritance,” in which offspring would be a mixture of the parents. Traits stayed distinct and could skip a generation. Mendel proposed that inheritance depends on units that come in pairs (now called genes with two alleles), that each parent passes on one unit of each pair, and that one form can hide the other.

3The vocabulary of inheritance

With complete dominance, a heterozygous individual (Bb) has the same phenotype as a homozygous dominant individual (BB). “Homozygous dominant and heterozygous organisms will look identical.” This is why a hidden recessive allele can pass silently through a generation, and why the white trait could reappear in the F₂.

4The law of segregation

Mendel’s first principle is the law of segregation: the two alleles of a gene separate (segregate) during the formation of gametes, so each gamete carries only one allele of each gene. Fertilization then restores the pair. The physical basis is meiosis. In the first meiotic division, homologous chromosomes are pulled apart, and with them the two alleles of each gene. An individual with the genotype Bb produces two kinds of gametes, half with B and half with b.

In the F₁ plants of Mendel’s cross (all Bb), each plant makes B and b gametes in equal numbers. Random fertilization of those gametes gives the F₂ a mix of genotypes, and a mix of phenotypes in a 3:1 ratio.

5Using a Punnett square

A Punnett square is a grid that shows all possible combinations of gametes from two parents. The steps are:

1. Write the genotype of each parent.
2. List the gametes each parent can make (one allele each).
3. Put one parent’s gametes along the top and the other’s down the side.
4. Fill each box with the allele from its column and the allele from its row.
5. Count the genotypes and phenotypes and convert them to ratios or probabilities.

For the cross Bb × Bb, the genotype ratio is 1 BB : 2 Bb : 1 bb, and the phenotype ratio is 3 dominant : 1 recessive. Each box has the same chance (one in four) because each gamete type is equally likely and fertilization is random.

The test cross

An organism with a dominant phenotype may be homozygous (BB) or heterozygous (Bb), and it is impossible to tell by looking. The solution is a test cross: the unknown individual is crossed with a homozygous recessive individual (bb). If all offspring show the dominant trait, the unknown parent is homozygous dominant. If the offspring appear in a 1:1 ratio of dominant to recessive, the unknown parent is heterozygous. Animal and plant breeders use this method to find out which individuals carry a hidden recessive allele.

6Probability: why ratios are not guarantees

A Punnett square gives probabilities, not promises. Each fertilization is an independent event, like a coin toss. A couple whose chance of having a child with a recessive condition is 1 in 4 can have four children with none affected, or two affected children in a row. Ratios only become close to the predicted values in large samples, which is why Mendel counted thousands of plants.

Two rules make it possible to calculate the chance of combined events.

• Product rule. The chance that two independent events both occur is the product of their separate chances.
• Sum rule. The chance that either one of two mutually exclusive events occurs is the sum of their chances.

7Two traits at once: the dihybrid cross and independent assortment

Mendel also followed two traits together, such as seed colour and seed shape. He crossed plants that were heterozygous for both traits (YyRr × YyRr). Each parent can produce four types of gametes (YR, Yr, yR and yr), so the Punnett square has 16 boxes, and the phenotypes appear in the ratio 9 : 3 : 3 : 1.

In one experiment with tall and inflated-pod peas, the counts were 2,706 tall and inflated, 930 tall and constricted, 888 dwarf and inflated and 300 dwarf and constricted, which is close to 9:3:3:1. The same result follows from multiplying probabilities: the chance of round and yellow is 3/4 × 3/4 = 9/16, and the chance of wrinkled and green is 1/4 × 1/4 = 1/16.

This outcome illustrates Mendel’s second principle, the law of independent assortment: alleles of different genes sort into gametes independently of each other. The physical basis is that homologous pairs line up independently of one another in the first division of meiosis. There is an important limit: genes that lie close together on the same chromosome tend to be inherited together (they are linked), and this departs from independent assortment. Crossing over can still separate them, and the less often it happens, the closer the genes are.

8Mendelian inheritance in human conditions

Many human traits and disorders follow Mendel’s patterns. A recessive condition appears only in people who inherit two altered copies, and a person with one altered copy is a carrier, who usually shows no symptoms but can pass the altered copy on. Autosomal dominant conditions, such as Huntington’s disease and Marfan syndrome, need only one altered copy.

• Cystic fibrosis is recessive. When both parents are carriers, each child has a 25 percent chance of having cystic fibrosis, a 50 percent chance of being a carrier, and a 25 percent chance of neither.
• Sickle cell disease is also recessive. When both parents are carriers, each child has a 1 in 4 chance of inheriting no sickle cell gene, a 1 in 2 chance of being a carrier, and a 1 in 4 chance of the disease.
• PKU, covered in the previous lesson, is recessive as well, and parents are usually carriers who are unaware of it.

Not every pattern is this simple. The next lesson covers incomplete dominance, co-dominance, multiple alleles and sex-linked inheritance, and shows how pedigrees are used to trace human traits through families.

9Practice problems with solutions

Problem 1: reading offspring ratios

A plant with purple flowers (dominant, P) is crossed with a plant with white flowers. Of 80 offspring, 41 are purple and 39 are white. What is the genotype of the purple parent?

Solution. The white parent must be pp. Offspring are about half purple and half white, which is the 1:1 result of a test cross with a heterozygote. The purple parent is Pp. If it were PP, all 80 offspring would be purple.

Problem 2: probability of several children

Two carriers of sickle cell disease plan three children. What is the chance that all three have the disease? Each child has a 1/4 chance, and the births are independent, so the product rule gives 1/4 × 1/4 × 1/4 = 1/64. The chance that none has the disease is (3/4)³ = 27/64.

Problem 3: a dihybrid shortcut

In a cross of YyRr × YyRr, what fraction of offspring are wrinkled and yellow (rr and at least one Y)? The chance of wrinkled is 1/4 and the chance of yellow is 3/4, so the product is 3/16, matching the 3 in the 9:3:3:1 ratio.

Summary

• Alleles separate into different gametes (segregation) and genes on different chromosomes sort independently.
• With complete dominance, heterozygous and homozygous dominant individuals look the same, so a test cross is needed to tell them apart.
• Punnett squares give probabilities for each birth, not guarantees for a small family.

?Quick check

Try each question first, then reveal the answer.