Chapter 5.6, Problem 6E

Population Genetics-First Cousins This is a continuation of Exercise $5$. Double first cousins are first cousins in two ways—that is, each of the parents of one is a sibling of a parent of the other. First cousins in general have six different grandparents whereas double first cousins have only four different grandparents. For mating between double first cousins, the proportions of the genotypes of the children can be accounted for in terms of those of parents, grandparents, and great-grandparent:

${\lambda }^3=\frac{1}{2}{\lambda }^2+\frac{1}{4}\lambda +\frac{1}{8}$.

a. Find all solutions to the cubic equation above and Identify which is ${\lambda }_1$.

b. After $5$ generations, what proportion of the population will be homozygous?

c. After $20$ generations, what proportion of the population will be homozygous?

$5$. Population Genetics 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 population in which blood-related individual mate, them 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 mating of blood-related parents is greater than that for children from unrelated parents

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 mating 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 previous two generations (represented by $\lambda$ for the parents’ generation and by the constant term of the grandparents’ generation).

a Find both solutions to the quadratic equation above and identify which is ${\lambda }_1$ (use a horizontal span of $-1$ to $1$ in this exercise and the following exercise.)

b After $5$ generations, what proportion of the population will be homozygous?

c After $20$ generations, what proportion of the population will be homozygous?