Question

The ability to taste the compound PTC is controlled by a dominant allele \(T,\) while individuals homozygous for the recessive allele \(t\) are unable to taste PTC. In a genetics class of 125 students, 88 can taste \(\mathrm{PTC}\) and 37 cannot. Calculate the frequency of the \(T\) and \(t\) alleles and the frequency of the genotypes in this population.

Short answer

Answer: In this population of 125 students, the frequency of the alleles and genotypes are as follows: - The frequency of the dominant allele T (p): 1-√(37/125) - The frequency of the recessive allele t (q): √(37/125) - The frequency of the homozygous dominant genotype TT: (1-√(37/125))^2 - The frequency of the heterozygous genotype Tt: 2(1-√(37/125))√(37/125) - The frequency of the homozygous recessive genotype tt: 37/125

Step-by-step solution

Calculate the frequency of each genotype

Since individuals homozygous for the recessive allele \(t\) are unable to taste PTC, the frequency of \(tt\) genotype can be directly calculated by dividing the number of non-taster individuals by the total population size: \(tt\) frequency = \(\frac{37}{125}\) Those who can taste PTC must either be homozygous dominant \(TT\) or heterozygous \(Tt\). As we don't have direct data, we will move to the next step to find the frequency of dominant and recessive alleles. Then, we can determine the frequency of \(TT\) and \(Tt\).

Calculate the frequency of alleles (p and q)

Let \(p\) be the frequency of the dominant allele \(T\), and let \(q\) be the frequency of the recessive allele \(t\). Since this is a diploid organism and has two alleles per individual, we know: \(p + q = 1\) Since individuals homozygous for the recessive allele \(t\) are unable to taste PTC, we can write: \(qq = tt = \frac{37}{125}\) Using this equation, we can find the frequency of the recessive allele \(t\): \(q=\sqrt{\frac{37}{125}}\) Now, we can find the frequency of the dominant allele \(T\): \(p = 1 - q = 1 - \sqrt{\frac{37}{125}}\)

Calculate the frequency of homozygous dominant and heterozygous genotypes

As we found the allele frequencies, \(p\) and \(q\), we can calculate the genotype frequencies. The frequency of homozygous dominant \(TT\) is given by: \(TT = p^2\) The frequency of heterozygous \(Tt\) is given by: \(Tt = 2pq\) Now we can plug in the values of \(p\) and \(q\) we found: \(TT = (1-\sqrt{\frac{37}{125}})^2\) \(Tt = 2(1-\sqrt{\frac{37}{125}})\sqrt{\frac{37}{125}}\)

Calculate the exact values of genotype frequencies

Now we can calculate the exact values of genotype frequencies: \(tt = \frac{37}{125}\) \(TT = (1-\sqrt{\frac{37}{125}})^2\) \(Tt = 2(1-\sqrt{\frac{37}{125}})\sqrt{\frac{37}{125}}\)

Conclusion

In this population of 125 students, the frequency of the alleles and genotypes are as follows: - The frequency of the dominant allele \(T (p)\) : \(1-\sqrt{\frac{37}{125}}\) - The frequency of the recessive allele \(t (q)\) : \(\sqrt{\frac{37}{125}}\) - The frequency of the homozygous dominant genotype \(TT\) : \((1-\sqrt{\frac{37}{125}})^2\) - The frequency of the heterozygous genotype \(Tt\) : \(2(1-\sqrt{\frac{37}{125}})\sqrt{\frac{37}{125}}\) - The frequency of the homozygous recessive genotype \(tt\) : \(\frac{37}{125}\)

Key concepts

Hardy-Weinberg Equilibrium

The Hardy-Weinberg equilibrium is a foundational concept in population genetics that provides a model for understanding genetic variation in a population under certain conditions. It establishes that allele and genotype frequencies in a large, randomly-mating population will remain constant over generations, provided there is no mutation, natural selection, gene flow, or genetic drift. It's expressed through the equation:
\( p^2 + 2pq + q^2 = 1 \),
where \( p \) and \( q \) represent the frequencies of two alleles of a genetic locus. The Hardy-Weinberg equilibrium serves as a null hypothesis for the study of evolutionary processes. In the context of PTC taste sensitivity, this principle allows us to predict that if the population is in Hardy-Weinberg equilibrium, the allele and genotype frequencies will not change from one generation to the next, providing a stable backdrop to investigate genetic patterns.

Allele Frequency

Allele frequency refers to how common an allele is in a population. It is a measure of genetic diversity and is represented by a proportion between 0 and 1 or a percentage. In the genotype of an individual organism, which consists of two alleles, the sum of the frequencies of both alleles will always add up to 1 (100%).

Calculating the allele frequency involves counting the number of times an allele appears within the population's gene pool and dividing it by the total number of alleles for a given gene. The PTC tasting ability example illustrates how to compute the allele frequency for the dominant \(T\) and the recessive \(t\) alleles using the Hardy-Weinberg principle. As the problem demonstrates, even when the genotype ratio (homozygous/heterozygous) is unknown, you can still deduce the allele frequency by analysis of the known genotype for the recessive, non-tasting trait.

Genotype Frequency

Genotype frequency is the proportion of a particular genotype within a population. It provides insight into the population's genetic structure and can be indicative of evolutionary changes over time. In a scenario like PTC taste sensitivity, determining genotype frequency involves analysis of how many individuals possess particular combinations of alleles.

By employing the Hardy-Weinberg equation, once we have the allele frequencies (\(p\) for the taster \(T\) allele and \(q\) for the non-taster \(t\) allele), we can easily calculate the expected genotype frequencies under equilibrium conditions: \(p^2\) signifies the proportion of homozygous dominant (taster), \(q^2\) for homozygous recessive (non-taster), and \(2pq\) for the heterozygous (taster). These predictive tools are valuable for researchers and educators in understanding the inheritance patterns within a population.

Dominant and Recessive Alleles

In the genetic context, alleles are considered dominant or recessive depending on their phenotypic expressions. A dominant allele is one that expresses its trait even when only one copy is present, while a recessive allele requires two copies (being homozygous) to exhibit its trait. The PTC taste sensitivity case showcases this concept with the \(T\) allele being dominant (tasters) and the \(t\) allele being recessive (non-tasters).

An individual with at least one dominant allele (\(TT\) or \(Tt\)) will be a taster, while only those individuals with two copies of the recessive allele (\(tt\)) will be non-tasters. Understanding these principles helps with predicting phenotypic ratios and is a fundamental aspect of studying heredity and genetic variation.