Chapter 20, Problem 8P

Summary Introduction

a.

To determine:

Whether the population is at Hardy-Weinberg equilibrium with respect to Q and R genes.

Introduction:

Geoffrey H. Hardy was a scientist who proposed the concept of Hardy-Weinberg equilibrium. This concept is used to associate the allele frequency with the genotype frequency. The populations that have allele frequency and the genotypic frequency at equilibrium follow the concept of Hardy-Weinberg equilibrium.

Explanation of Solution

The given information is as follows:

$\text{Total population = 1480}$

The frequency of all the genotypes is as follows:

No.	Genotypes	Frequency
1.	QFQF RCRC	$\begin{array}{rll}f& =& \frac{202}{1480}\\ & =& 0.137\end{array}$
2.	QFQG RCRC	$\begin{array}{rll}f& =& \frac{101}{1480}\\ & =& 0.068\end{array}$
3.	QGQG RCRC	$\begin{array}{rll}f& =& \frac{101}{1480}\\ & =& 0.068\end{array}$
4.	QFQF RCRD	$\begin{array}{rll}f& =& \frac{372}{1480}\\ & =& 0.251\end{array}$
5.	QFQG RCRD	$\begin{array}{rll}f& =& \frac{186}{1480}\\ & =& 0.126\end{array}$
6.	QGQG RCRD	$\begin{array}{rll}f& =& \frac{186}{1480}\\ & =& 0.126\end{array}$
7.	QFQF RDRD	$\begin{array}{rll}f& =& \frac{166}{1480}\\ & =& 0.112\end{array}$
8.	QFQG RCRD	$\begin{array}{rll}f& =& \frac{83}{1480}\\ & =& 0.056\end{array}$
9.	QGQG RCRD	$\begin{array}{rll}f& =& \frac{83}{1480}\\ & =& 0.056\end{array}$

$\begin{array}{rll}\text{Frequency of }{Q}^F{Q}^F& =& \text{0}\text{.137}{Q}^F{Q}^F{R}^C{R}^C+\text{0}\text{.251}{Q}^F{Q}^F{R}^C{R}^D+\text{0}\text{.112}{Q}^F{Q}^F{R}^D{R}^D\\ & =& \text{0}\text{.5}\end{array}$

$\begin{array}{rll}\text{Frequency of }{Q}^F{Q}^G& =& \text{0}\text{.068}{Q}^F{Q}^G{R}^C{R}^C+\text{0}\text{.126}{Q}^F{Q}^G{R}^C{R}^D+\text{0}\text{.056}{Q}^F{Q}^G{R}^C{R}^D\\ & =& \text{0}\text{.25}\end{array}$

$\begin{array}{rll}\text{Frequency of }{Q}^G{Q}^G& =& \text{0}\text{.068 }{Q}^G{Q}^G{R}^C{R}^C+\text{0}\text{.126 }{Q}^G{Q}^G{R}^C{R}^D+\text{0}\text{.056 }{Q}^G{Q}^G{R}^C{R}^D\\ & =& \text{0}\text{.25}\end{array}$

$\begin{array}{rll}\text{Frequency of }Q\text{gene}& =& \text{0}\text{.137 }{Q}^F{Q}^F{R}^C{R}^C+\text{0}\text{.251 }{Q}^F{Q}^F{R}^C{R}^D+\text{0}\text{.112 }{Q}^F{Q}^F{R}^D{R}^D\\ & =& \text{0}\text{.5 }{Q}^F{Q}^F+\text{0}\text{.25 }{Q}^F{Q}^G+\text{0}\text{.25}{Q}^G{Q}^G\\ & =& 1\end{array}$

$\begin{array}{rll}\text{Allele frequency of }{Q}^F& =& \text{0}\text{.625}\\ \text{Allele frequency of }{Q}^G& =& \text{0}\text{.375}\end{array}$

$\begin{array}{rll}\text{Expected genotype frequencies in the next generation}& =& {\left(0.625\right)}^2{Q}^F{Q}^F+\left(2\right)\left(0.625\right)\left(0.375\right)Q^F{Q}^G+{\left(0.375\right)}^2{Q}^G{Q}^G\\ & =& \text{0}\text{.39 }{Q}^F{Q}^F+\text{0}\text{.47 }{Q}^F{Q}^G+\text{0}\text{.14 }{Q}^G{Q}^G\\ & =& \text{1}\end{array}$

The observed genotype frequencies of Q gene are not close to the expected genotype frequencies. This indicates that the population is not at equilibrium for Q gene.

$\begin{array}{rll}\text{Frequency of}{R}^C{R}^C& =& \text{0}\text{.137 }{Q}^F{Q}^F{R}^C{R}^C+\text{0}\text{.068}{Q}^F{Q}^G{R}^C{R}^C+\text{0}\text{.068 }{Q}^G{Q}^G{R}^C{R}^C\\ & =& \text{0}\text{.273}\end{array}$

$\begin{array}{rll}\text{Frequency of }{R}^C{R}^D& =& \text{0}\text{.251 }{Q}^F{Q}^F{R}^C{R}^D+\text{0}\text{.126 }{Q}^F{Q}^G{R}^C{R}^D+\text{0}\text{.126 }{Q}^G{Q}^G{R}^C{R}^D\\ & =& \text{0}\text{.5}\end{array}$

$\begin{array}{rll}\text{Frequency of }{R}^D{R}^D& =& \text{0}\text{.112}{Q}^F{Q}^F{R}^D{R}^D+\text{0}\text{.056 }{Q}^F{Q}^G{R}^C{R}^D+\text{0}\text{.056}{Q}^G{Q}^G{R}^C{R}^D\\ & =& \text{0}\text{.22}\end{array}$

$\begin{array}{rll}\text{Frequency of }R\text{ gene}& =& \text{0}\text{.273 }{R}^C{R}^C+\text{0}\text{.5 }{R}^C{R}^D+\text{0}\text{.22 }{R}^D{R}^D\\ & =& \text{1}\end{array}$

$\begin{array}{rll}\text{Allele frequency of }{R}^C& =& \text{0}\text{.52}\\ \text{Allele frequency of }{R}^D& =& \text{0}\text{.48}\end{array}$

$\begin{array}{rll}\text{Expected genotype frequencies in the next generation}& =& {\left(0.52\right)}^2{R}^C{R}^C+\left(2\right)\left(0.52\right)\left(0.48\right)+{\left(0.48\right)}^2{R}^D{R}^D\\ & =& \text{0}\text{.27 }{R}^C{R}^C+\text{0}\text{.5 }{R}^C{R}^D+\text{0}\text{.23 }{R}^D{R}^D\\ & =& 1\end{array}$

The observed genotype frequencies of R gene are close to the expected genotype frequencies. This reflects that the population is at equilibrium for R gene.

Thus, the population is at Hardy-Weinberg equilibrium with respect to R gene.

Summary Introduction

b.

To determine:

The fraction of population in the next generation that will be Q^F Q^F.

Introduction:

Q^F and Q^G are two alleles of gene Q. This gene codes for a particular type of red blood cells. The gene Q plays a crucial role in a particular blood grouping type.

Explanation of Solution

The given information is as follows:

No.	Genotypes	Frequency
1.	QFQF RCRC	$\begin{array}{rll}f& =& \frac{202}{1480}\\ & =& 0.137\end{array}$
2.	QFQF RCRD	$\begin{array}{rll}f& =& \frac{372}{1480}\\ & =& 0.251\end{array}$
3.	QFQF RDRD	$\begin{array}{rll}f& =& \frac{166}{1480}\\ & =& 0.112\end{array}$

$\begin{array}{rll}\text{Frequency of }{Q}^F{Q}^F& =& \text{0}\text{.137}{Q}^F{Q}^F{R}^C{R}^C+\text{0}\text{.251}{Q}^F{Q}^F{R}^C{R}^D+\text{0}\text{.112}{Q}^F{Q}^F{R}^D{R}^D\\ & =& \text{0}\text{.5}\end{array}$

$\text{Allele frequency of }{Q}^F=\text{0}\text{.625}$

$\begin{array}{rll}\text{Expected genotype frequency of}{Q}^F\text{ in the next generation}& =& {\left(0.625\right)}^2{Q}^F{Q}^F\\ & =& \text{0}\text{.39 }{Q}^F{Q}^F\end{array}$

Thus, the fraction of population in the next generation that will be Q^F Q^F is 0.39.

Summary Introduction

c.

To determine:

The fraction of population in the next generation that will be R^C R^C.

Introduction:

R^C and R^D are two alleles of gene R. This gene also codes for a particular type of red blood cells. However, the blood group typing of R gene is different from the blood group typing of the Q gene.

Explanation of Solution

The given information is as follows:

No.	Genotypes	Frequency
1.	QFQF RCRC	$\begin{array}{rll}f& =& \frac{202}{1480}\\ & =& 0.137\end{array}$
2.	QFQG RCRC	$\begin{array}{rll}f& =& \frac{101}{1480}\\ & =& 0.068\end{array}$
3.	QGQG RCRC	$\begin{array}{rll}f& =& \frac{101}{1480}\\ & =& 0.068\end{array}$

$\begin{array}{rll}\text{Frequency of}{R}^C{R}^C& =& \text{0}\text{.137 }{Q}^F{Q}^F{R}^C{R}^C+\text{0}\text{.068}{Q}^F{Q}^G{R}^C{R}^C+\text{0}\text{.068 }{Q}^G{Q}^G{R}^C{R}^C\\ & =& \text{0}\text{.273}\end{array}$

$\text{Allele frequency of }{R}^C=\text{0}\text{.52}$

$\begin{array}{rll}\text{Expected genotype frequency of}{R}^C\text{ in the next generation}& =& {\left(0.52\right)}^2{R}^C{R}^C\\ & =& \text{0}\text{.27 }{R}^C{R}^C\end{array}$

Thus, the fraction of the population in the next generation that will be R^C R^C is 0.27.

Summary Introduction

d.

To determine:

The chance that the first child of a Q^C Q^G R^C R^D female and a Q^F Q^F R^C R^D male will be Q^F Q^G R^D R^D male.

Introduction

The process by which a male fertilizes with a female to produce an offspring is termed as reproduction. This is an important process to maintain the population of an organism in the environment. The chances that the offspring would be a male or a female depend upon both the parents.

Explanation of Solution

$\begin{array}{rll}\text{Chances that female would contribute}{Q}^G\text{ allele}& =& \frac{\text{1}}{\text{2}}\\ \text{Chances that male would contribute }{Q}^F\text{ allele}& =& \frac{\text{1}}{\text{1}}\\ \text{Chances that childwould have }{Q}^F{Q}^G\text{allele}& =& \frac{\text{1}}{\text{2}}\end{array}$

The parents are homozygous for the R gene. This reflects that:

$\begin{array}{rll}\text{Chances that child would be homozygous }{R}^D{R}^D& =& \frac{\text{1}}{\text{4}}\\ \text{Chances that child would be a male}& =& \frac{\text{1}}{\text{2}}\end{array}$

The overall probability that the child would be Q^F Q^G R^D R^D male can be calculated as:

$\begin{array}{rll}\text{Overall probability}& =& \left(\frac{\text{1}}{\text{2}}\right)\times \left(\frac{\text{1}}{\text{4}}\right)\times \left(\frac{\text{1}}{\text{2}}\right)\\ & =& \frac{\text{1}}{\text{16}}\end{array}$

Thus, the chance that the first child of a Q^C Q^G R^C R^D female and a Q^F Q^F R^C R^D male will be Q^F Q^G R^D R^D male is 1/16.