Question

If 4 percent of a population in equilibrium expresses a recessive trait, what is the probability that the offspring of two individuals who do not express the trait will express it?

Short answer

Answer: The probability is approximately 2.56%.

Step-by-step solution

Determine the frequency of the recessive allele

Use the given frequency of the recessive trait (4%) to determine the frequency of the recessive allele (q) in the population. Since individuals with the recessive trait have two copies of the recessive allele, we know that q^2 = 0.04. To find the frequency of the recessive allele, take the square root of 0.04: q = sqrt(0.04) = 0.2

Find the frequency of the dominant allele

Now that we have the frequency of the recessive allele (q), we can find the frequency of the dominant allele (p) using the equation p + q = 1: p = 1 - q = 1 - 0.2 = 0.8

Determine the genotype frequencies of the parents

The two non-expressing individuals must have the genotype Aa or AA. The genotype frequencies can be found using their respective Hardy-Weinberg equations: Frequency of Aa: 2pq = 2 * 0.8 * 0.2 = 0.32 Frequency of AA: p^2 = (0.8)^2 = 0.64

Calculate the probability of each parental combination

Since there are two possible genotypes for each non-expressing parent, there are four possible combinations of parents. Using the genotype frequencies determined in Step 3, we can calculate the probability of each combination: AA x AA: 0.64 * 0.64 = 0.4096 AA x Aa: 0.64 * 0.32 = 0.2048 Aa x AA: 0.32 * 0.64 = 0.2048 Aa x Aa: 0.32 * 0.32 = 0.1024

Determine the probability of an offspring expressing the recessive trait

In each parental combination, calculate the probability of producing an offspring with the recessive genotype (aa): AA x AA: 0% chance (no recessive alleles) AA x Aa: 0% chance (offspring will either be AA or Aa) Aa x AA: 0% chance (offspring will either be AA or Aa) Aa x Aa: 25% chance (1 out of 4 possible combinations result in aa) Now, sum the probabilities weighted by the parental probabilities: 0.4096 * 0% + 0.2048 * 0% + 0.2048 * 0% + 0.1024 * 25% = 0.0256 The probability that the offspring of two individuals who do not express the trait will express it is approximately 2.56%.

Key concepts

Genotype Frequency

Understanding genotype frequency is crucial when studying genetic variation within a population. Genotype frequency refers to how often a specific genetic makeup occurs in a population. It is usually expressed as a proportion or a percentage. For example, in a population, if we want to find out the frequency of a dominant homozygous genotype (AA), we need to determine the proportion of individuals that are AA out of the total population.

In the case of the Hardy-Weinberg equilibrium, genotype frequencies are predicted by the equations p² for the frequency of AA, 2pq for Aa, and q² for aa, where p is the frequency of the dominant allele, and q is the frequency of the recessive allele. It is imperative to recognize that the Hardy-Weinberg principle assumes random mating, no mutation, no migration, large population size, and no selection to illustrate ideal conditions for these frequencies to remain constant from generation to generation.

To calculate these frequencies accurately, one must first identify p and q, and then use these to find the proportion of each genotype. When it comes to solving problems like the one provided, being comfortable with determining genotype frequencies is an essential skill, allowing students to make precise predictions about genetic traits.

Recessive Allele Frequency

The concept of recessive allele frequency is another cornerstone of population genetics. It is defined as the proportion of all alleles in the population that are the recessive allele, represented by q in Hardy-Weinberg equations. A recessive allele is one that can be masked by the presence of a dominant allele. Therefore, its expression in a phenotype is only visible when an individual has two copies of this allele (is homozygous recessive).

In the provided exercise, starting with the expression of a recessive trait, which is given as 4% or 0.04, we were asked to calculate the frequency of the recessive allele. To do so, we assume Hardy-Weinberg equilibrium and use the principle that the frequency of the homozygous recessive genotype (aa) is equal to q². By taking the square root of the expression frequency, we find q, giving us the recessive allele frequency within the population. With this frequency, we can proceed to tackle more complex problems involving predicting outcomes of genetic crosses, understanding patterns of inheritance, and determining allele frequencies in future generations.

Probability of Genetic Trait Expression

When it comes to understanding the probability of genetic trait expression, it is critical to grasp the laws of inheritance and how they apply in a population context. The probability that an offspring will express a genetic trait depends on the genotypes of the parents and the mode of inheritance of the trait (dominant or recessive).

In our exercise, we deal with a recessive trait. To express a recessive trait, the offspring must inherit two copies of the recessive allele, which is symbolized as 'aa'. Using the Hardy-Weinberg formulae, we calculate the probability of the parents producing an 'aa' genotype offspring. This problem takes into account that the parents do not express the trait (meaning they are either AA or Aa) and combines the probabilities of each possible mating scenario to find the overall probability that an offspring expresses the recessive trait.

This calculation is fundamental to genetic counseling, research in genetic disorders, and understanding how traits are passed on through generations. By integrating the Hardy-Weinberg principle with Mendelian genetics, we can predict such probabilities with surprising accuracy, providing valuable insights into the expression of genes within a population.