Problem 29 - Algebra and Trigonometry

Question

Many genetic traits are controlled by two genes, one dominant and one recessive. In Gregor Mendel’s original experiments with peas, the genes controlling the height of the plant are denoted by T (tall) and t (short). The gene T is dominant, so a plant with the genotype (genetic makeup) TT or Tt is tall, whereas one with genotype tt is short. By a statistical analysis of the offspring in his experiments, Mendel concluded that offspring inherit one gene from each parent, and each possible combination of the two genes is equally likely. If each parent has the genotype Tt, then the following chart gives the possible genotypes of the offspring: (Table not Copy) Find the probability that a given offspring of these parents will be (a) tall or (b) short.

Step-by-Step Solution

Answer

Probability of tall offspring is 0.75; probability of short offspring is 0.25.

Step 1: Understanding the Genotypes

In Mendel's experiment, each parent has a genotype of Tt. The dominant gene T results in being tall, while the recessive gene t results in being short. Therefore, offspring need TT or Tt to be tall and tt to be short.

Step 2: Constructing the Punnett Square

The parent's alleles can be combined in the following ways using the Punnett square:\[\begin{array}{c|cc} & T & t \\hlineT & TT & Tt \t & Tt & tt \\end{array}\]This square shows all possible combinations of offspring genotypes.

Step 3: Determining the Probability for Tall Offspring

From the Punnett square, the following combinations are possible: TT, Tt, Tt, tt. The genotypes TT and Tt result in tall offspring. So, there are 3 tall genotypes out of 4 possible combinations.

Step 4: Calculating the Probability for Tall Offspring

Probability of a tall offspring = \(\frac{\text{Number of tall genotypes}}{\text{Total number of possible combinations}} = \frac{3}{4} = 0.75\).

Step 5: Determining the Probability for Short Offspring

The genotype tt results in short offspring. From the Punnett square, there is only 1 tt genotype out of the 4 possible combinations.

Step 6: Calculating the Probability for Short Offspring

Probability of a short offspring = \(\frac{\text{Number of short genotypes}}{\text{Total number of possible combinations}} = \frac{1}{4} = 0.25\).

Key Concepts

Punnett Square

A Punnett Square is a useful tool in genetics that helps predict the genetic makeup of offspring from a specific mating scenario. It was named after Reginald Punnett, a pioneer in the field of genetics. This simple grid allows us to map out all possible combinations of parental genes and visually represents the probability of an offspring inheriting particular traits.

To construct a Punnett Square, list the alleles (gene versions) from one parent on the top and the alleles from the other parent along the side. For example, if each parent has the genotype Tt, which corresponds to a tall and short gene, you would write one parent's T and t along one side and the other parent's T and t along the top.

• Fill in the squares by combining the alleles from each row and column intersection.
• This results in four possible combinations of genotypes—TT, Tt, Tt, and tt in our example.

The Punnett Square not only helps us see the potential genetic variations in the offspring but also aids in calculating their probabilities.

Genotype

The term 'genotype' refers to the genetic constitution of an organism, particularly in terms of the specific alleles or gene versions it carries. Understanding genotypes is crucial in genetics because they determine the physical traits or phenotypes of an organism. In Mendelian genetics, a genotype is often discussed in terms of alleles being homozygous or heterozygous.

• Homozygous genotypes: These have two identical alleles, either TT (homozygous dominant) or tt (homozygous recessive).
• Heterozygous genotypes: These have two different alleles, such as Tt, where one allele is dominant and the other is recessive.

In Mendel's pea plant experiments, a genotype of TT or Tt resulted in tall plants because the T allele is dominant. Conversely, a genotype of tt resulted in short plants because both alleles were recessive. These genotype combinations represent the genetic blueprint that influences how traits are expressed in living organisms.

Dominant and Recessive Traits

In genetics, traits are often classified as either dominant or recessive based on how they are expressed. Dominant traits are those that can mask the effect of other traits, while recessive traits can be hidden by a dominant one. Understanding these concepts is key to predicting the inheritance patterns in Mendelian genetics.

• Dominant Traits: These require only one copy of the allele to be expressed. In the pea plant example, the tall trait (T) is dominant, which means that plants with the genotypes TT or Tt will both appear tall.
• Recessive Traits: These need two copies of the recessive allele to be visibly expressed. Hence, a plant will only be short with the genotype tt.

Dominance and recessiveness help explain how certain characteristics appear in offspring. Dominant alleles often mask the presence of recessive ones, but the recessive traits can still be passed to the next generation and might reappear if two recessive alleles are inherited.