Unit 3 Population Genetics: Gene frequencies and Hardy Weinberg Equilibrium

Introduction to Population Genetics

• Population: A group of individuals of the same species occupying a given area that can freely interbreed and produce fertile offspring
• Gene pool: The sum total of genes of all individuals in a population, consisting of all alleles at all gene loci

Allele and Genotype Frequencies

• Allele frequency (a.k.a. Gene frequency): Proportion of an allele in the gene pool compared to other alleles at the same locus
  • p = frequency of dominant allele (A), q = frequency of recessive allele (a)
  • p + q = 1
• Genotype frequency: Proportion of individuals with a specific genotype in the population
  • AA genotype frequency = p²
  • Aa genotype frequency = 2pq
  • aa genotype frequency = q²

In a population with one-locus-two-alleles condition (A and a), three kinds of individuals will occur – AA, Aa, aa.

• Therefore: If – N = Total no. of individuals in the population

         D = No. of dominant homozygous individuals

         H = No. of heterozygous individuals

         R = No. of homozygous recessive individuals

Then, AA genotype frequency = D/N

           Aa genotype frequency = H/N

           aa genotype frequency = R/N

• genotype frequency is obtained by dividing the number of individuals with that genotype by the total number of individuals in the population.

• Sample problem: In a population of 100 individuals with 40 homozygous dominant (AA), 40 heterozygous (Aa), and 20 homozygous recessive (aa) individuals, then allele frequencies (using the allele counting method) will be as follows:
  • Frequency of A = ((2×40) + 40)/200 = 0.6
  • Frequency of a = (40+40)/200 = 0.4

Hardy-Weinberg Equilibrium

• States that allele and genotype frequencies remain constant over generations in a large, randomly mating population when evolutionary forces (migration, mutation, selection) are absent.
• It describes the situation where the population is undergoing no evolutionary change
• Assumptions:
  • No mutation, selection, migration
  • Large population size
  • Random mating
  • Equal gamete production and random combination
• Genotype frequencies: p² (AA) + 2pq (Aa) + q² (aa) = 1

Factors Affecting Hardy-Weinberg Equilibrium

• Small population size: Genetic drift causes random fluctuations in allele frequencies
• Non-random mating: Inbreeding and assortative mating change genotype frequencies
• Mutation: Introduces new alleles, usually at low rates
• Migration: Gene flow between populations alters allele frequencies
• Selection: Different genotypes have different fitness, changing allele frequencies
  • Types: Directional, balancing, disruptive

Applications of HW Law

1. Calculation of frequencies of dominant and recessive genes in a population
   1. Recessive homozygotes are phenotypically distinguishable, hence frequency can be calculated. Frequency of heterozygotes and dominant homozygotes can not be estimated based on phenotype aloneIf population is in HW eq^m, knowledge of the frequency of the dominant allele can be estimated using that of the recessive allele, as p + q = 1
2. Calculation of frequencies of carriers or heterozygotes in a population
   • Can be estimated if the gene frequency of one allele is known
   • 1. Use: freq = 2pq = 2p(1-p) = 2(1-q)q
   • 1. Important – to figure out the frequency of carriers of recessive genes or abnormalities in a population
3. To test for agreement with a population in HW eq^m –  (to test whether a population is evolving)
   1. If gene frequencies are available – expected frequency of genotypes are calculated
   1. Observed genotype frequency is estimated from the population
   1. Using chi-square test on both observed and expected frequencies, significance of any discrepancy observed is tested

Example:

1. In a population of 1000 cats, coat color is determined by a single gene with two alleles. The dominant allele B produces black coat color, while the recessive allele b produces white coat color. After surveying the population, researchers found 490 black cats and 510 white cats.

Find the following:

1. Calculate the frequencies of the B and b alleles in this population.
   1. Using the calculated allele frequencies, predict the expected genotype frequencies (BB, Bb, bb) in the next generation, assuming Hardy-Weinberg equilibrium.

Solutions:

Calculating allele frequencies:

Frequency of b allele (q): √(510/1000) = √0.51 = 0.714

Frequency of B allele (p): 1 – 0.714 = 0.286

Predicting genotype frequencies:

BB: p² = 0.286^2 = 0.082 or 8.2%

Bb: 2pq = 2(0.286)(0.714) = 0.408 or 40.8%

bb: q² = 0.714^2 = 0.51 or 51%

1. In a population of 500 cats, a recessive allele causes a genetic disorder. After testing, 36 cats were found to have the disorder. Calculate:

a) The frequency of the recessive allele (q)
b) The frequency of the dominant allele (p)
c) The expected number of heterozygous carriers in the population

Solution:
a) Calculating the frequency of the recessive allele (q):

• Number of affected cats (homozygous recessive) = 36
• Total population = 500
• Frequency of affected individuals = 36/500 = 0.072
• Since q^2 = 0.072 (frequency of homozygous recessive)
• q = √0.072 = 0.268 (rounded to three decimal places)

b) Calculating the frequency of the dominant allele (p):

• Since p + q = 1
• p = 1 – q = 1 – 0.268 = 0.732

c) Calculating the expected number of heterozygous carriers:

• Frequency of heterozygotes = 2pq
• 2pq = 2 × 0.732 × 0.268 = 0.392
• Expected number of heterozygotes = 0.392 × 500 = 196 cats

Therefore:
a) The frequency of the recessive allele (q) is 0.268 or 26.8%
b) The frequency of the dominant allele (p) is 0.732 or 73.2%
c) The expected number of heterozygous carriers in the population is 196 cats

Max. gene frequency of any allele

p=q=0.5

Changes in Gene Frequencies Due to Migration and Mutation

Migration (Gene Flow)

Migration introduces new alleles into a population or changes existing allele frequencies. The effect depends on:

• Migration rate (m): Proportion of population composed of migrants
• Allele frequency difference between migrant and resident populations

Change in allele frequency due to migration:

Δp = m(p_m – p)

Where:
Δp = Change in allele frequency
m = Migration rate
p_m = Allele frequency in migrant population
p = Initial allele frequency in resident population

Sample Problem:

A population has an allele frequency of 0.6 for allele A. Migrants with an allele frequency of 0.8 for A enter the population at a rate of 0.1 per generation. Calculate the new allele frequency after one generation of migration.

Solution:

Δp = m(pm – p)

= 0.1(0.8 – 0.6)

= 0.1(0.2)

= 0.02

New allele frequency = p + Δp = 0.6 + 0.02 = 0.62

Mutation

Mutation introduces new alleles or changes existing ones. The effect depends on:

• Mutation rate (μ): Probability of a gene mutating per generation
• Back mutation rate (ν): Probability of mutant allele reverting to original form

Change in allele frequency due to mutation:

Δp = ν(1-p) – μp

Where:
Δp = Change in allele frequency
ν = Back mutation rate
μ = Forward mutation rate
p = Initial allele frequency

Sample Problems:

1. In a population, allele A mutates to allele a at a rate of 1 × 10^-5 per generation. The back mutation rate from a to A is 2 × 10^-5 per generation. If the initial frequency of A is 0.8, calculate the change in allele frequency due to mutation in one generation.

Solution:
Δp = ν(1-p) – μp
= (2 × 10^-5)(1-0.8) – (1 × 10^-5)(0.8)
= (2 × 10^-5)(0.2) – (8 × 10^-6)
= 4 × 10^-6 – 8 × 10^-6
= -4 × 10^-6

The frequency of allele A will decrease by 0.000004 in one generation due to mutation.

2. A population has an initial frequency of 0.7 for allele B. It experiences migration from a population with a B frequency of 0.9 at a rate of 0.05 per generation. Additionally, B mutates to b at a rate of 2 × 10^-6 per generation, with no back mutation. Calculate the new allele frequency after one generation.

Solution:
Change due to migration: Δp_m = m(p_m – p) = 0.05(0.9 – 0.7) = 0.01
Change due to mutation: Δpμ = -μp = -(2 × 10^-6)(0.7) = -1.4 × 10^-6

Total change: Δp = Δpm + Δpμ = 0.01 – 0.0000014 = 0.0099986

New allele frequency = 0.7 + 0.0099986 = 0.7099986

Factors Affecting HW Eq^m

1. Mutations

• Definition: Sudden, inheritable changes in genetic material
• Types and Examples:
  • Point mutations: Single nucleotide change (e.g., sickle cell anemia in humans)
  • Chromosomal mutations: Large-scale changes (e.g., Down syndrome in humans)
• Effects:
  • Introduce new alleles into the population
  • Can be beneficial, neutral, or deleterious
• Example: In cattle, a mutation in the myostatin gene leads to double-muscling, affecting meat production

2. Gene Flow (Migration)

• Definition: Transfer of genetic variation between populations due to movement of individuals or gametes
• Effects:
  • Alters allele frequencies in both source and recipient populations
  • Can introduce new alleles or change existing allele frequencies
• Example: Introduction of polled (hornless) cattle breeds into horned populations, altering the frequency of the polled allele

3. Genetic Drift

• Definition: Random changes in allele frequencies due to chance events
• Types and Examples:
  • Bottleneck effect: Drastic reduction in population size (e.g., near-extinction of American bison)
  • Founder effect: Establishment of a new population by a small number of individuals (e.g., Chillingham cattle in England)
• Effects:
  • More pronounced in small populations
  • Can lead to loss of genetic variation
• Example: Loss of genetic diversity in small, isolated populations of endangered breeds like the Florida Cracker cattle

4. Non-random Mating

• Definition: Mating patterns that deviate from random mating assumptions
• Types and Examples:
  • Inbreeding: Mating between closely related individuals (e.g., line breeding in pedigree dogs)
  • Assortative mating: Mating between individuals with similar phenotypes (e.g., selection for coat color in cattle)
• Effects:
  • Increases homozygosity in the population
  • Can lead to deviations from Hardy-Weinberg proportions
• Example: Increased incidence of genetic disorders in highly inbred dog breeds, such as hip dysplasia in German Shepherds

5. Natural Selection

• Definition: Differential survival and reproduction of individuals based on their genetic makeup
• Types and Examples:
  • Directional selection: Favors one extreme of a trait (e.g., artificial selection for milk yield in dairy cattle)
  • Stabilizing selection: Favors intermediate phenotypes (e.g., birth weight in most mammals)
  • Disruptive selection: Favors extreme phenotypes (e.g., beak size in Galápagos finches)
• Effects:
  • Changes allele frequencies over time
  • Can lead to adaptation and evolution

Example: Development of antibiotic resistance in bacteria affecting livestock, such as Staphylococcus aureus in dairy cows