Problem 54 - Calculus Early Transcendentals

Question

Three alleles (alternative versions of a gene) \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{O}\) determine the four blood types \(\mathrm{A}(\mathrm{AA}\) or \(\mathrm{AO}), \mathrm{B}(\mathrm{BB}\) or \(\mathrm{BO})\) \(\mathrm{O}(\mathrm{OO}),\) and \(\mathrm{AB} .\) The Hardy-Weinberg Law states that the pro- portion of individuals in a population who carry two different alleles is $$P=2 p q+2 p r+2 r q$$ where \(p, q,\) and \(r\) are the proportions of \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{O}\) in the population. Use the fact that \(p+q+r=1\) to show that \(P\) is at most \(\frac{2}{3} .\)

Step-by-Step Solution

Answer

The maximum value of \( P \) is \( \frac{2}{3} \).

Step 1: Write the Expression for P

Given the Hardy-Weinberg Law, start with the expression for \( P \), which is \( P = 2pq + 2pr + 2rq \).

Step 2: Substitute using the Constraint

Substitute \( r = 1 - p - q \) into the expression for \( P \) using the constraint \( p + q + r = 1 \). This gives: \[ P = 2pq + 2p(1 - p - q) + 2(1 - p - q)q \].

Step 3: Simplify the Expression

Expand and simplify the expression for \( P \): \[ P = 2pq + 2p(1 - p - q) + 2(1 - p - q)q = 2pq + 2p - 2p^2 - 2pq + 2q - 2pq - 2q^2 \].

Step 4: Simplify Further

Simplify further: \[ P = 2p + 2q - 2p^2 - 4pq - 2q^2 \].

Step 5: Find Maximum Value

To find the maximum, make \( 2p(1-p-q) + 2q(1-q-p) \). Use symmetry to consider values like \( p = q = r = \frac{1}{3} \), giving: \[ P = 2pq + 2pr + 2rq = 2\times\frac{1}{3}\times\frac{1}{3} + 2\times\frac{1}{3}\times\frac{1}{3} + 2\times\frac{1}{3}\times\frac{1}{3} = 2 \times \frac{1}{9} + 2 \times \frac{1}{9} + 2 \times \frac{1}{9} = \frac{6}{9} = \frac{2}{3} \].

Step 6: Verify the Maximum using Tools

The calculation using symmetry and substitution shows that \( P \) achieves maximum value at \( \frac{2}{3} \) when \( p, q, \text{and } r \) are each \( \frac{1}{3} \). Any deviation from equal distribution among alleles leads to a lesser \( P \).

Key Concepts

blood type genetics

The understanding of blood type genetics is essential as it explains the inheritance of different blood types. Humans have four main blood types: A, B, AB, and O. These blood types are determined by the presence or absence of specific antigens, which are proteins or molecules, on the surface of red blood cells. The genetic basis for these blood types lies in the combination of three alleles: A, B, and O.

• The A allele codes for the A antigen.
• The B allele codes for the B antigen.
• The O allele, being recessive, does not code for any antigen.

Thus, individuals with type A blood may have AA or AO genotypes, type B may be BB or BO, type AB must be AB, and type O must be OO. This simple Mendelian inheritance allows for the predictable passing of blood types from parents to children, determined by the combination of alleles (one from each parent).

allele frequency

Allele frequency refers to how often an allele occurs in a population. In the context of blood type genetics, it tells us the proportion of alleles A, B, and O in the entire population. Understanding allele frequencies is crucial as it helps predict the genetic diversity within a population and assists in solving problems related to Hardy-Weinberg equilibrium. If you were to calculate allele frequency, you would need to count how many times the A, B, and O alleles appear relative to all alleles present in the population. For instance:

• If allele A is present in half the population, its frequency would be 0.5 or 50%.
• If allele B is in 30% of the population, its frequency would be 0.3.
• Consequently, allele O should be in 20% of the population, given the frequencies must add up to 1.

Tracking these frequencies is particularly useful in understanding how a population's genetic makeup changes over time, whether due to natural selection, genetic drift, or mutation.

Hardy-Weinberg equilibrium

The Hardy-Weinberg equilibrium is a principle that provides a mathematical baseline for understanding genetic variation in a population that is not undergoing evolution. In its simplest form, it allows us to predict the frequency of genotypes from the frequency of alleles. According to the Hardy-Weinberg principle, allele and genotype frequencies in a population will remain constant from generation to generation if certain conditions are met:

• The population is infinitely large.
• Mating is random.
• No mutations occur.
• No immigration (gene flow).
• No natural selection occurs.

In the context of our exercise, the Hardy-Weinberg equilibrium provides the foundation for deriving the formula for P, which calculates the proportion of individuals carrying two different alleles. By applying this principle, you can explain genetic distributions and understand whether or not a real population is evolving.

probability of genotype

Probability of genotype is a concept that describes the likelihood of a specific genetic makeup occurring in an individual within a population. When dealing with blood type genetics, you are primarily interested in determining how likely it is for a person to have a particular combination of alleles (like AA, AO, BB, etc.). Calculating these probabilities involves using allele frequencies. For instance:

• The chance of inheriting an AA genotype is computed as the square of the frequency of the A allele (\(p^2\)).
• The probability for AO is 2 times the product of the A allele frequency and the O allele frequency (\(2pr\)).
• Similar calculations apply to BB (\(q^2\)) and BO (\(2qr\)) genotypes.

These probabilities help us to understand the distribution of blood types in a population and to make predictions about genetic inheritance patterns. Through the lens of the Hardy-Weinberg principle, these probabilities become significant tools in genetic research and studies of population genetics.