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Population Genetics

Genotype and Allele Frequency Estimation is the first step in studying a polymorphism. Used for family data and independent individuals in a population. We can use a subset of individuals who are independent and count alleles, or use the maximum likelihood methods to take all genotypes into account for pedigree data.

Consider the following example of allele frequency estimation:image-1664839258179.png
We can take the frequency of each allele by the observed proportions:
    pM = (2*298 + 489)/(2*1000)
    pN = 1 - pM

Hardy-Weinberg Law

Describes how we expect allele frequencies and phenotype frequencies to be related in a population.

  1. For a large, random-mating population, in the absence of forces that change allele frequencies, the allele and genotype frequencies remain constant from one generation to the next
  2. After one generation of random mating, for an autosomal locus with alleles 1 and 2 (frequencies p and q = 1 - p), the relative frequencies of the genotypes 11, 12, and 22 are:
         p2, 2pqm q2

image-1664839771200.png

Assumptions
  • Random mating with respect to genotype
    • No assortative mating
    • No population structure
  • No selection, mutation, or migration
  • Discrete generations
  • Infinite population size
  • Autosomal locus

Given two alleles with 1 and 2, there are 6 possible parent mating types:image-1664840002929.png

So the frequency of allele 1 in the offspring would be:
    P(11) + ½P(12) = (u+v/2)2 + (u+v/2)(v/2+w) = (u+v/2)(u+v+w) = u+v/2

Similarly, the frequency of allele 2 is: w + v/2

Forces that change allele or genotype frequency (invalidate HW law)
  • Mutation
  • Migration
  • Selection
    • Deleterious mutations tend to be rare if there is selection against them
      • Exception: Heterozygote advantage for a recessive deleterious
  • Drift - small populations
  • Non-random mating
Testing for HWE

Though several assumptions of the HW law are not met in any population, genotypes in a population usually conform reasonable well to expectations, due to the various forces cancelling each other out.

H0: The genotype frequencies math the HW expectations (p2, 2pqm q2)

  1. Estimate allele frequency (p_hat)
  2. Determine the expected genotype frequencies from the estimated allele frequency, assuming null is true

    image-1664840626013.png

  3. Compute the expeceted genotype counts

    image-1664840650695.png

  4. Compare observed genotype counted to expected
    image-1664840698879.png

Compare X2 to a chi-squared distribution with 1 degree of freedom. This is usually the number of categories minus 1, but we lose an additional degree of freedom since we estimate allelic frequencies from the data (3-1-1).

Conclusion: The observed genotype frequencies are [not] significantly different from the expectations of the HW equilibrium.

When we reject the HWE, we usually don't know why other than the assumptions being violated.

Exact HWE Test

There are (2N)!/nA!nB! possible arrangements for the alleles in the sample. Under HWE the probability of observing exactly nAB heterozygotes in N individuals with nA alleles is:image-1664841170550.png

Under many conditions, samples of affected individuals will not be in HWE for alleles associated with disease BUT controls should be close to HWE, as should population-based (unascertained) samples. Note also that genotypes among related individuals may not be in HWE since the individuals are not independent.

 

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