Hardy-Weinberg Equilibrium and Sex Linkage: Principles and Applications in Genetics

Hardy-Weinberg Equilibrium

Definition and Conditions

The Hardy-Weinberg Equilibrium (HWE) describes a theoretical state in which allele and genotype frequencies in a large, randomly mating population remain constant from generation to generation, provided that certain conditions are met. These conditions include:

• Large population size

• Random mating

• No migration

• No mutation

• No selection

When these criteria are satisfied, the population is said to be in Hardy-Weinberg Equilibrium, and genotype frequencies are determined solely by allele frequencies.

Mathematical Formulation

• Let p be the frequency of allele A, and q be the frequency of allele a (p + q = 1).

• Genotype frequencies are given by:

  • Frequency of AA: $p^2$

  • Frequency of Aa: $2pq$

  • Frequency of aa: $q^2$

Relative Frequencies of Alleles and Genotypes

Allele frequencies can be calculated from genotype counts:

• $p = P + \frac{1}{2}H$

• $q = Q + \frac{1}{2}H$

• Where P = frequency of AA, Q = frequency of aa, H = frequency of Aa

Random Mating and Genotype Frequencies

Random mating ensures that allele frequencies do not change from one generation to the next. The genotype frequencies after random mating are:

• $f(AA) = p^2$

• $f(Aa) = 2pq$

• $f(aa) = q^2$

Computing a Punnett Square is equivalent to calculating genotype frequencies under random mating.

Estimating Allele Frequencies

Allele frequencies can be estimated from observed genotype frequencies, especially for recessive traits:

• If the frequency of a recessive genotype (aa) is known, $q$ can be estimated as $q = \sqrt{f(aa)}$

• This estimation assumes Hardy-Weinberg Equilibrium and ignores the contribution of heterozygotes (Aa) to the allele frequency.

• If the frequency of heterozygotes is known, a more accurate estimate is: $q = Q + \frac{1}{2}H$

Applications: Detection of Carriers

Hardy-Weinberg principles are used to detect carriers of recessive alleles in populations:

• For example, in cattle, a Black Angus bull of unknown genotype (BB or Bb) can be test-mated with a bb Red Angus cow.

• If any offspring are bb (red), the bull must be a carrier (Bb).

• The probability of not detecting a carrier after n test crosses is $(1/2)^n$ if each cross is independent.

• General formula for detection after n crosses with various cow genotypes: $P(\text{Detect } Bb \text{ after } n) = 1 - [f(BB) + \frac{3}{4}f(Bb) + \frac{1}{2}f(bb)]^n$ where $f(\text{genotype})$ is the frequency of that genotype in test crosses.

Example Table: Relative Frequencies of Alleles and Genotypes

Genotype	Individuals	Allele A	Allele a
AA	30	60	0
Aa	60	60	60
aa	10	0	20
Total	100	120	80

Proportions: $P = 0.3$, $H = 0.6$, $Q = 0.1$

Sex Linkage and Hardy-Weinberg Equilibrium

Sex Linkage: Mammals vs Birds

Sex-linked inheritance involves genes located on sex chromosomes. The pattern of inheritance differs between mammals and birds:

• Mammals: Females are XX, males are XY. Males contribute either X or Y gametes; females contribute only X.

• Birds (Avian species): Females are ZW, males are ZZ. Males contribute only Z gametes; females contribute Z or W.

Sex Linkage in Mammals: Generational Changes

Allele frequencies in sex-linked genes change differently in males and females across generations:

• Let $p_m$ and $q_m$ be allele frequencies in males, $p_f$ and $q_f$ in females.

• After one generation, allele frequencies in males become those previously in females, and vice versa.

• Equilibrium is approached over several generations, with average allele frequency stabilizing.

Key Equations:

• Frequency in males in next generation: $p_m' = p_f$

• Frequency in females in next generation: $p_f' = \frac{1}{2}(p_m + p_f)$

• General average: $\bar{p} = \frac{2p_f + p_m}{3}$

Example Table: Sex Linkage Generational Changes

Generation	p_m	p_f	p_m - p_f	p̄
0	0.3	0.6	-0.3	0.5
1	0.6	0.45	0.15	0.5
2	0.45	0.525	-0.075	0.5

Example: Expected Frequencies from Sex-Linked Crosses

Given parental genotype frequencies, the expected frequencies of sons and daughters can be calculated using Punnett squares and allele frequency formulas.

• For example, if female genotypes are AA (0.36), Aa (0.48), aa (0.16) and male genotypes are A_ (0.5), a_ (0.5):

  • Sons: $p_M = 0.6$, $q_M = 0.4$

  • Daughters: $p_F = 0.55$, $q_F = 0.45$

Summary Table: Sex Linkage in Mammals (Parental Generation)

Genotype	Count	Allele Frequency	Gametes
Male A_	30	p_m = 0.3	0.3 A_, 0.7 a_
Male a_	70	q_m = 0.7
Female AA	30	p_f = 0.6	0.6 A, 0.4 a
Female Aa	60
Female aa	10	q_f = 0.4

Key Concepts and Applications

• Hardy-Weinberg Equilibrium provides a baseline for detecting evolutionary forces such as selection, mutation, and migration.

• Carrier detection is a practical application in animal breeding and genetic counseling.

• Sex linkage requires modified equilibrium calculations due to differences in inheritance patterns between sexes.

Additional info: These notes expand on the provided slides by including definitions, formulas, and structured tables for clarity and completeness.