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In the study of population genetics, an important measure of inbreeding is the proportion of homozygous genotypes-that is, instances in which the two alleles carried at a particular site on an individual's chromosomes are both the same. For populations in which bloodrelated individuals mate, there is a higher than expected frequency of homozygous individuals. Examples of such populations include endangered or rare species, selectively bred breeds, and isolated populations. In general, the frequency of homozygous children from matings of blood-related parents is greater than that for children from unrelated parents. \(^{67}\) Measured over a large number of generations, the proportion of heterozygous genotypes-that is, nonhomozygous genotypes-changes by a constant factor \(\lambda_{1}\) from generation to generation. The factor \(\lambda_{1}\) is a number between 0 and 1 . If \(\lambda_{1}=0.75\), for example, then the proportion of heterozygous individuals in the population decreases by \(25 \%\) in each generation. In this case, after 10 generations the proportion of heterozygous individuals in the population decreases by \(94.37 \%\), since \(0.75^{10}=\) \(0.0563\), or \(5.63 \%\). In other words, \(94.37 \%\) of the population is homozygous. For specific types of matings, the proportion of heterozygous genotypes can be related to that of previous generations and is found from an equation. For matings between siblings, \(\lambda_{1}\) can be determined as the largest value of \(\lambda\) for which $$ \lambda^{2}=\frac{1}{2} \lambda+\frac{1}{4}. $$ This equation comes from carefully accounting for the genotypes for the present generation (the \(\lambda^{2}\) term) in terms of those of the previous two generations (represented by \(\lambda\) for the parents' generation and by the constant term for the grandparents' generation). a. Find both solutions to the quadratic equation above and identify which is \(\lambda_{1}\). (Use a horizontal span from \(-1\) to 1 in this exercise and the following two exercises.) b. After 5 generations what proportion of the population will be homozygous? c. After 20 generations what proportion of the population will be homozygous?

Step by step solution

01

Solve the Quadratic Equation

Start by rewriting the quadratic equation \( \lambda^2 = \frac{1}{2} \lambda + \frac{1}{4} \) in standard form. This gives \( \lambda^2 - \frac{1}{2} \lambda - \frac{1}{4} = 0 \). To find the roots, use the quadratic formula: \[\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Substituting \( a = 1 \), \( b = -\frac{1}{2} \), and \( c = -\frac{1}{4} \), calculate the roots.

02

Apply the Quadratic Formula

Plug the coefficients into the quadratic formula: \[\lambda = \frac{-(-\frac{1}{2}) \pm \sqrt{(-\frac{1}{2})^2 - 4 \times 1 \times (-\frac{1}{4})}}{2 \times 1} = \frac{\frac{1}{2} \pm \sqrt{\frac{1}{4} + 1}}{2}\] Simplify inside the square root: \[\lambda = \frac{\frac{1}{2} \pm \sqrt{\frac{5}{4}}}{2} = \frac{\frac{1}{2} \pm \frac{\sqrt{5}}{2}}{2}\] This gives \( \lambda = \frac{1 \pm \sqrt{5}}{4} \). Solve for both \( \lambda_1 \) and the other root.

03

Identify \(\lambda_1\)

Given that we need \( \lambda_1 \) to be the largest value between \(-1\) and \(1\), the relevant solution is \( \lambda_1 = \frac{1 + \sqrt{5}}{4} \). This value is greater than \( \frac{1 - \sqrt{5}}{4} \), and it's between \(0\) and \(1\).

04

Compute Homozygous Proportion After 5 Generations

The problem states that the homozygous population increases with each generation as the heterozygous proportion decreases by a factor of \( \lambda_1 \). Given 5 generations and initial heterozygosity of \(100\%\), calculate: \[\text{Heterozygous proportion after 5 generations} = \lambda_1^5\]\[\text{Homozygous proportion} = 1 - \lambda_1^5\] Substitute \( \lambda_1 = \frac{1 + \sqrt{5}}{4} \) to find the numerical answer.

05

Compute Homozygous Proportion After 20 Generations

Using the same approach as in Step 4, compute the heterozygous proportion after 20 generations: \[\text{Heterozygous proportion after 20 generations} = \lambda_1^{20}\]\[\text{Homozygous proportion} = 1 - \lambda_1^{20}\] Again substitute \( \lambda_1 = \frac{1 + \sqrt{5}}{4} \) to find the numerical result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inbreeding

Inbreeding refers to the mating of closely related individuals within a population, leading to a higher frequency of homozygous genotypes. This happens because these individuals are more likely to inherit the same alleles from common ancestors. Inbreeding is common in certain populations such as those that are isolated, endangered, or selectively bred.

The main concern with inbreeding is the loss of genetic diversity. A high frequency of homozygous individuals can lead to the expression of deleterious recessive traits. These are traits that can result in reduced fitness, survival, and reproduction of individuals. Inbreeding might be seen as a negative factor in natural populations but could be advantageous in selective breeding where specific traits are desirable.

It's crucial to monitor inbreeding levels in conservation and breeding programs to ensure the long-term survival of species. Methods like pedigree analysis and genome-wide assessments help quantify inbreeding and potentially harmful allelic combinations.

Homozygous Genotypes

A genotype refers to the genetic makeup of an organism, particularly with regard to a particular trait or set of traits. Homozygous genotypes are when the two alleles at a given locus on a chromosome are identical.

Homozygosity is important because it can influence whether certain traits are expressed or suppressed. For example, in the case of a recessive disease allele, an individual will only express the disease if they are homozygous for that allele.

In genetics, homozygous genotypes can lead to traits becoming fixed within a population, particularly if inbreeding is prevalent. This may reduce genetic diversity over time. However, in some breeding scenarios, achieving homozygosity for specific alleles can help establish permanent traits within a strain or lineage.

Recognizing homozygosity is essential for understanding inheritance patterns, disease risk factors, and the genetic health of populations.

Quadratic Equation

Quadratic equations are fundamental in population genetics for modeling various phenomena. In this instance, the equation \( \lambda^2 = \frac{1}{2} \lambda + \frac{1}{4} \) is derived from genetic principles.

A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \). Solving it involves finding values of \( x \) that satisfy the equation. These solutions often reveal crucial properties about the system in question, such as the factor by which genetic traits are passed onto future generations.

The quadratic formula \[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] is used to solve these equations. By substituting the specific coefficients \( a = 1 \), \( b = -\frac{1}{2} \), and \( c = -\frac{1}{4} \), you can determine the values of \( \lambda \). In genetic problems, \( \lambda \) often represents a change factor in time, dictating how frequencies of genetic traits will be affected over generations.

Heterozygous Proportion

Heterozygous individuals have two different alleles at a particular locus. In population genetics, heterozygosity is a measure of genetic variation. It signifies the proportion of individuals carrying two different alleles for a particular trait.

The proportion of heterozygous individuals decreases in inbred populations, as shown by the factor \( \lambda_1 \). This factor illustrates the rate at which indicators of genetic diversity diminish each generation. If \( \lambda_1 \) is less than 1, the heterozygous proportion decreases over time. For instance, if \( \lambda_1 = 0.75 \), the heterozygous proportion decreases by 25% every generation.

Calculating the change in heterozygosity aids in predicting future genetic configurations of populations. This understanding helps in creating strategies for preserving genetic diversity, especially in small or isolated groups where genetic drift and inbreeding are pronounced. It is a critical index for ensuring population viability.