Question

Evolutionary fitness. Suppose that the probability of having the dominant allele (D) in a gene is \(p\) and the probability of the recessive allele \((\mathbf{R})\) is \(q=1-p\). You have two alleles, one from each parent. (a) Write the probabilities of all the possibilities: DD, DR, and RR. (b) If the fitness of \(\mathrm{DD}\) is \(f_{\mathrm{DD}}\), the fitness of \(\mathrm{DR}\) is \(f_{\mathrm{DR}}\), and the fitness of \(R R\) is \(f_{R R}\), write the average fitness in terms of \(p\).

Short answer

1. Probabilities: \(Pr(DD) = p^2, Pr(DR) = 2pq, Pr(RR) = (1-p)^2\)2. Average fitness: \overline{w} = p^2 \cdot f_{DD} + 2p(1-p) \cdot f_{DR} + (1-p)^2 \cdot f_{RR}\.

Step-by-step solution

- Identify Probabilities of Each Allele

Given that the probability of having the dominant allele (D) is denoted by \(p\), the probability of having the recessive allele (R) is \(q = 1 - p\).

- Calculate Probabilities of Each Genotype

We need to determine the probabilities for the combinations of alleles (DD, DR, RR) using the given probabilities \(p\) and \(q\). \[ Pr(DD) = p^2 \] \[ Pr(DR) = 2pq \] \[ Pr(RR) = q^2 \]

- Express Fitness Values for Each Genotype

We are given the fitness values for each genotype: \(f_{DD}\) for DD, \(f_{DR}\) for DR, and \(f_{RR}\) for RR.

- Compute the Average Fitness

The average fitness, denoted as \overline{w}\, is calculated by multiplying the fitness of each genotype by its corresponding probability and then summing them up: \[ \overline{w} = Pr(DD) \cdot f_{DD} + Pr(DR) \cdot f_{DR} + Pr(RR) \cdot f_{RR} \] Substitute the probabilities from Step 2: \[ \overline{w} = p^2 \cdot f_{DD} + 2pq \cdot f_{DR} + q^2 \cdot f_{RR} \] Since \(q = 1 - p\): \[ \overline{w} = p^2 \cdot f_{DD} + 2p(1-p) \cdot f_{DR} + (1-p)^2 \cdot f_{RR} \]

Key concepts

alleles

In evolutionary genetics, alleles are different versions of a gene. Each individual inherits two alleles for each gene, one from each parent. For a particular gene, an allele can be dominant or recessive.
Dominant alleles are typically represented by a capital letter (E.g., D), while recessive alleles are denoted with a lowercase letter (E.g., r).
The combination of these alleles determines the genotype of the organism.

genotype probabilities

Genotype probabilities refer to the likelihood of an individual having a particular combination of alleles. In our example, if the probability of having a dominant allele D is denoted by p, then the probability of having a recessive allele r is q, where q = 1 - p.
To determine the probabilities of the different genotype combinations (DD, DR, RR), we use the following formulas:

• Probability of DD: \(Pr(DD) = p^2\)
• Probability of DR: \(Pr(DR) = 2pq\)
• Probability of RR: \(Pr(RR) = q^2\)

These probabilities help us predict the distribution of genotypes in the population.

fitness values

Fitness values measure how well a particular genotype reproduces and survives in an environment. These values are essential in understanding natural selection.
In our example, we have three genotypes with the following fitness values:

• Fitness of DD: \(f_{DD}\)
• Fitness of DR: \(f_{DR}\)
• Fitness of RR: \(f_{RR}\)

Higher fitness values mean greater success in passing genes to the next generation.

average fitness

Average fitness, denoted as \overline{w}\, represents the overall reproductive success of a population's gene pool.
To calculate it, we consider the fitness values and the genotype probabilities. The formula for average fitness is:
\[ \overline{w} = Pr(DD) \cdot f_{DD} + Pr(DR) \cdot f_{DR} + Pr(RR) \cdot f_{RR} \] By substituting the genotype probabilities, we get: \[ \overline{w} = p² \cdot f_{DD} + 2pq \cdot f_{DR} + q² \cdot f_{RR} \] Given that q = 1 - p, this can be further simplified to: \[ \overline{w} = p² \cdot f_{DD} + 2p(1 - p) \cdot f_{DR} + (1 - p)² \cdot f_{RR} \]
This equation considers the contributions of each genotype to the overall fitness of the population.

dominant and recessive alleles

Understanding dominant and recessive alleles is fundamental in genetics.
A dominant allele (D) expresses its trait even if only one copy is present, while a recessive allele (r) requires two copies to display its trait.
For example, if an individual has the DR genotype, the dominant allele D will mask the effect of the recessive allele r.
However, if both alleles are recessive (RR), the recessive trait will be expressed. This distinction helps explain how certain traits are inherited and how they might provide an evolutionary advantage or disadvantage.