Hardy-Weinberg Equilibrium & Allele Frequencies

Allele Frequency

Refers to how frequently a particular allele is detected (or observed) in a population

Calculating Allele Frequency: Two Methods

Assumptions:

• We are dealing with a single locus in a large and random mating population

• We are dealing with diploid species

• Can calculate allele frequency by counting or proportions

Calculating Allele Frequency by Counting

Simply add the number of alleles in each genotype

• Ex: in the A1A1 genotype, there’s 78 individuals, so the count of the A1 allele in that genotype is 78 + 78 = 156

Notations for Allele Frequency

In the context of genotypes and allele frequency: the dominant allele is the most common allele in that population

• Most common allele frequency is p

  • In this case, A1

• Minor allele frequency (MAF) is q

  • In this case, A2

Finding frequency (see image):

• f(A1) = 210/300 = 0.7 (this is p)

• f(A2) = 90/300 = 0.3 (this is q)

Calculating Allele Frequencies by Proportion

Genotype frequencies: P, H, and Q

• P is the frequency of homozygous A1 (A1A1)

• Q is the frequency of homozygous A2 (A2A2)

• H is the frequency of heterozygous genotype (A1A2)

When calculating, simply divide the number of individuals of that genotype out of the total number of individuals

• Ex: P = f(A1A1) = 78/300 = 0.52

To find the frequency of A1 (p) or A2 (q), you use:

• p = P + ½ H

• q = Q + ½ H

Rules & Assumptions for Calculating Allele Frequencies

• p = P + ½ H

• q = Q + ½ H

• P + H + Q = 1

  • P = 1 - (H + Q)

  • Q = 1 - (H + P)

• p + q = 1

  • p = 1 - q

  • q = 1 - p

• Assumptions: large number and random mating (AKA hard-weinberg)

Hardy-Weinberg Equilibrium

Occurs in a large and random mating population in the absence of migration, mutation, and selection

• Allele frequencies and genotypic frequencies remain constant

• Genotypic frequencies are determined by allele frequencies

Comparing Frequencies Between Generations

If you compare the genotypic and allelic frequencies in g₀ and g₁ and they are not constant, that means the g₀ population is not in HWE

• Only one generation of random mating is required for g₀ population to reach HWE

Estimating Allele Frequencies Using Recessive Condition Genotypes

ONLY use f(bb) to estimate f(b) if f(Bb) is not available. Since we’re ignoring the contribution of heterozygotes to the q calculation, we’re assuming the population is in HWE

• P = p²

• H = 2pq

• Q = q²

We know there’s 18 recessive individuals (bb), so we can find Q

• Q = f(bb) → 18/200 = 0.09

• If Q = q², then take the square root of Q (AKA q²) to get q: √0.09 = 0.3

• Now we can find p: 1 - q = p → 1 - 0.3 = 0.7

Example: Est. Allele Frequencies Using Recessive Condition Genotypes

If you are told 6.0% of the population suffers from a recessive condition, what is the frequency of the dominant allele (rounded to three decimal places)? What is the assumption to calculate the recessive allele frequency?

• f(aa) = 0.06

• Assume HWE

• q = √Q = √q² → √0.06 = 0.24

• p = 1 - 0.24 = 0.76

Genotype Frequency in the Next Generation

• f(BB) = p²

• f(Bb) = 2pq

  • This is the carrier

• f(bb) = q²

  • The percentage (frequency) of the individual progeny (the next generation of individuals) showing the genetic condition

Properties of Equilibrium Populations

1. Frequency of heterozygote: H = 2pq

• The biggest H (H_max) can be is 0.5

2. H = 2√PQ or H/√PQ = 2

• Provides a quick test for determine HWE without needing to use allele frequencies

• If it is equal to 2, then the population is in HWE

Detection of Carriers

Carriers are individuals of any species that carries a single allele of a genetic variant (i.e. heterozygote) that is associated with a specific genetic disorder. Can detect carriers by:

• DNA Testing Technology

  • SNP chip that tests for several genetic conditions or disorders

  • Relatively cheap and provides lots of services

• Test mating

Use of HWE in Detection of Carriers

Example: we have a locus controlling coat color with genotypes: BB, Bb, and bb

• What’s the probability of being a carrier? (Bb)

• P(Bb) = the probability of being carrier or having a red calf

• Level of confidence: there is an X% chance that we’d detect the bull is a carrier

Use of HWE in Detection of Carriers: Test Mating

P(detect Bb) = 1 - P(detect Bb after n)

• General rule: P(detect Bb after n) = 1 - [1*f(BB) + ¾ f(Bb) + ½ f(bb)]ⁿ

• When n is the number of progeny

• The 1, ¾ and ½ represent the respective probability of not seeing a red calf

Example: Detection of Carriers

P(detect Bb after n) = 1 - [1*f(BB) + ¾ f(Bb) + ½ f(bb)]ⁿ

• If all 1000 cows are BB, f(BB) = 1

  • Level of confidence = P(detect Bb) → 1-1¹⁰⁰⁰ = 1 - 1 = 0

  • 100% of the cows will be black

• If mated to 10 cows, ½ are Bb and ½ are bb

  • P(detect Bb) = 1 - [ ¾ (½) + ½(½)]¹⁰ = 0.9909 or 99%

    • We’re 99% confident that a cow is not a carrier

  • We know none of these cows are BB, so we don’t include that factor in the equation

Application of HWE

Genotypes Quality Control (QC) for genomic studies

• If genotypes aren’t in HWE, they’re excluded before being used in further analyses as they’re considered genotyping errors

Assumption: loci not in HWE can be lethal or causing disease

• Use HWE for deception of lethal alleles is hard because of the need to use a large number of individuals to detect rare variants and standard QC eliminates these genotypes

HWE in Testings

• Parentage testing (humane + livestock)

• Forensics (wildlife + crime)

  • In these test panels, the markers (SNPs) should be in HWE and segregate in most of the breeds

  • These tests are very sensitive to genotyping errors

HWE in the Real World

Most populations in the real world may not be in HWE, but HWE serves as a baseline model for studying the effects of many evolutionary and demographic factors on the genetic structure of the population

• I.e. the departure from HWE

• Genetic and genomic improvement have been done using selection or crossbreeding, so if we maintain all conditions for HWE, no genetic improvement would be achieved

Conditions of HWE

• A random mating population

• A large population

• No migration

• No mutation

• No selection

• If any of these aren’t met, HWE is violated

  • Ex: no random mating would be assortative mating and autosomal and sex linkage

  • Ex: migration would be crossing/importing and exporting semen and embryos

Random Mating

• A mating system in which all matings are equally likely

• To achieve HWE from a non-equilibrium population, one generation of random mating is required

Violating Random Mating: Positive Assortative Mating

• Individuals with similar phenotypes or genotypes mating with each other

• Mating like to like

  • Based on pedigree likeness or individual likeness such as performance success, disposition, body shape, etc.

• Leads to more homozygotes

• Ex: purebreeding

Violating Random Mating: Negative Assortative Mating

• Unlike mates to unlike (i.e. different genotypes or phenotypes mating together)

• Leads to more heterozygotes

• Ex: crossbreeding

(+) or (-): change in genotype frequency but not allele frequency

Recap of Assortative Matings

Assortative matings with no selection lead to change in genotype frequencies not allele frequencies (no removal; we aren’t doing selection)

• (+) assortative matings: more to homozygotes like purebreeding

• (-) assortative matings: more to heterozygotes like crossbreeding

Factors Affecting HWE: Selection

Goal of selection in a breeding program:

• To remove deleterious alleles

• To increase frequencies of favorable genes (alleles) in a population

• Selection for or against a particular allele changes in both genotypic and allele frequencies

Selection Under Heterozygote Advantage

There’s an overdominance for fitness (i.e. selection favors Aa genotype

• For any initial allele frequency, the population converges on a max heterozygosity (H = 0.5)

Selection Under Heterozygote Disadvantage

There’s an underdominance for fitness (i.e. selection acts against Aa genotype)

• Starting p < 0.5 pops head toward loss

• Starting p > 0.5 pops head towards fixation

• Populations converge at minimum heterozygosity

Selection Against a Recessive Homozygote

Selections acts against the aa genotype

• Frequency of dominant allele (A) rapidly approaches fixation from any initial allele frequency

Factors Affecting HWE: Migration

 

Migration is movement of alleles from one population to another

• Animals don’t necessarily need to move but if their genetic materials moved using reproductive technology (e.g. frozen embryos, semen)

• If allele frequencies are the same between the two populations, then the migration has no effect on allele frequency in the new population

• If allele frequencies are different, then migration alter the allele frequency for the new pop.

Factors Affecting HWE: Small Population

• Sampling from a small population will lead to random genetic drift

• Genetic drift is the random change in allele frequencies due to sampling a finite number of parents