Lecture 41: Population Genetics I:

Variation, Hardy-Weinberg, and Linkage Disequilibrium

(version 12 November 1999)

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• Population genetics: Deals directly with genotypes

• Quantitative genetics: Deals with phenotypes and make inferences about the underlying genetics

Ideally, we would be able to look at phenotypic variation and directly infer the underlying amounts of genetic variation. However, we cannot do this.

• 1800's - 1900's: Variation between the phenotypes of domestic breeds (Darwin)
  • The consistence differences between breeds was taken as evidence of genetic differences
  • Unclear if the variation within breeds was due to genetic or environmental variation (or both)

• 1900's - 1940's: detection of visible mutants in natural populations (mostly human genetic diseases)

• 1930's - 1950's: chromosomal inversion polymorphisms in Drosophila (Dobzhansky)

• 1960's: variation in protein sequence as measured by charge changes detected by starch-gel electrophoresis (Lewontin and Hubby). The electrophoretically-detectable variants are referred to as allozyme variants
  • The so-call find-em and grind-em approach to measuring biochemical variation as scored through allozyme variation
  • Extensive allozyme variation was found in almost all populations
    • a few exceptions, such as the California elephant seals, which show essentially no allozymic variation

• late 1980's-present: Indirect and direct measures of DNA variation
  • RFLP, restriction fragment length polymorphisms
  • RAPDs, randomly amplified polymorphic DNAs
  • Direct sequencing of alleles at a locus
    • One of the 1st studies looked that the Drosophila ADH locus (Krietman, 1983)
    • Extensive polymorphism found in most studies, typically about 1/1000 nucleotides is polymorphic
    • For example, you and your neighbor differ at (on average) 22,000,000 nucleotide pairs.

• Historical measures of variation (from the electrophoretic days where only a few allozyme variants were detected per locus)
  • Heterozygosity: What fraction of individuals are heterozygotes for the particular locus being examined

  • Fraction of polymorphic loci : what fraction of loci measured show polymorphism (two or more alleles at reasonable frequencies)

• the more current measure is the fraction of polymorphic sites -- What fraction of base pairs at a locus shows variation in the population? Typical values are about 1/500 to 1/1000.

Obtaining allele frequencies from genotype frequencies

• Freq(A₁) = Freq(A₁ A₁ homozygotes) + (1/2) Freq(all A₁ heterozygotes)

Predicting genotype frequencies from allele frequencies

1. Random mating (individuals mate independent of their genotype)
2. No selection (all genotypes leave, on average, the same number of offspring)
3. Large population size (genetic drift can be ignored)
4. Allele frequencies the same in both sexes
5. Autosomal loci

Then

• Freq(AA) = Freq(A from father)*Freq(A from mother) = Freq(A)*Freq(A)
  • The frequency of a homozygote is the square of the allele frequency

• Freq(Aa) = Freq(A from father)*Freq(a from mother) + Freq(a from father)*Freq(A from mother) = 2* Freq(A)*Freq(a)
  • The frequency of a heterozygote is twice the product of allele frequencies

• Further, the allele frequencies remain unchanged each generation (Homework problem!!)

What happens when the sexes have different allele frequencies? (Homework problem!!)

When considering two (or more) loci, one must also account for the presence ot Linkage disequilibrium

Under random mating, gametes combine at random. Hence, if the frequency of (say) an AB gamete is 0.4 and an ab gamete is 0.1, then

• freq(AB/AB) = 0.4²
• freq(ab/ab) = 0.1²
• freq(AB/ab) = 2*0.4*0.1

However, the frequencies of gametes in a population can change by recombination from generation to generation unless they are in linkage equilibrium (which is also called gametic-phase equilibrium).

Example

At linkage equilibrium, the gametes have frequencies expected by independence of alleles,

• i.e. Freq(AB gamete) = freq(A)*freq(B)

If linkage disequilibrium exists, how does it change over time under random mating?

Let D_AB(t) denote the disequilibrium for a particular gamete type, where

• D_AB(t) = Freq(AB in generation t) - Freq(A)*Freq(B)

( We also use the notation p_AB = Freq(AB), p_A = Freq(A), etc.)

Rearranging gives

• Freq(AB in gen. t+1) - Freq(A)*Freq(B) = (1-c) [ Freq(AB in gen. t) - Freq(A)*Freq(B) ]

Hence, the recursion for linkage disequilibrium is

• D(t+1) = D(t)*(1-c)
• so that D(t) = D(0)*(1-c)^t

Thus, for a particular gamete type, say AB (A and B represent particular alleles at two different loci), then

• D_AB(0) = Freq(AB in the initial generation) - Freq(A)*Freq(B)
• D_AB(t) = [ Freq(AB in the generation t) - Freq(A)*Freq(B) ] = D_AB(0)(1-c)^t
• Putting these together, Freq(AB in the generation t)
  • = Freq(A)*Freq(B) + [Freq(AB in the initial generation) - Freq(A)*Freq(B)]
  • = Freq(A)*Freq(B) + D_AB(0)(1-c)^t

Consider the Ainu population. Does this show indications of linkage disequilibrium?

• freq(M) = freq(MS) + freq(Ms) = 0.024 + 0.381 = 0.405
• freq(N) = 1 - freq(M) = 1 - 0.405= 0.595
• freq(S) = freq(MS) + freq(NS) = 0.024 + 0.247 = 0.271
• freq(s) = 1 - freq(S) = 1 - 0.271 = 0.729

Is this true? freq(MS gamete) = 0.024, while freq(M)*freq(S) = 0.405*0.271 = 0.110

Hence, the initial disequilibrium for the MS gamete is

• D_MS(0) = 0.024 - 0.110 = -0.086

If random mating and other Hardy-Weinberg assumptions hold, if the M-S locus distance is c = 0.1, what is the expected MS equilibrium after one generation of recombination?

• D_MS(1) = (1-c)* D_MS(0) = 0.9*(-0.086) = -0.0774
• Hence, Freq(MS in gen 1) = freq(M)*freq(S) + D_MS(1) = 0.110 -0.0774 = 0.0326

What about after 20 generations?

• D_MS(20) = (1-c)²⁰* D_MS(0) = 0.9²⁰*(-0.086) = -0.0774 = -0.010
• Hence, Freq(MS in gen 20) = freq(M)*freq(S) + D_MS(20) = 0.110 -0.010 = 0.0995

Most genes show linkage-disequilibrium between very tightly-linked markers

The usefulness of this observation is that we can use tightly-linked markers as indicators of whether a particular chromosome carries a disease allele.

How does this association arise?

Even after hundreds of generations, most chromosomes carrying the mutant allele also contain the tightly linked alleles on the original chromosome.

This feature has been exploited for very fine mapping of diseases genes, an approach called linkage-disequilibrium mapping

For any mutant alleles, there is a predominant haplotype (collection of very tightly-linked markers) with which it is associated, reflecting the haplotype of the original chromosome on which the mutant arose.

Equating the probability of no recombination to the observed proportion q of disease-bearing chromosomes with this predominant haplotype gives q= (1-c)^t, where t is the age of the mutation or the age of the founding population (whichever is more recent). Solving for recombination frequency gives

• c = 1-q^(1/t)

Hastbacka et al. (1992) examined the gene for diastrophic dysplasis (DTD), an autosomal recessive disease, in Finland.

A number of marker loci were examined, with the CSF1R locus showing the most striking correlation with DTD. The investigators were able to unambiguously determine the haplotypes of 152 DTD-bearing chromosomes and 123 normal chromosomes for the sampled individuals. Four alleles of the CSF1R marker gene were detected.

Here, q = 0.947, while the current Finnish population traces back to around 2000 years to a small group of founders, which underwent around t=100 generations of exponential growth.

Using these estimates of q and t, gives an estimated recombination frequency between the CSF1R gene and the DTD gene as

c = 1- (0.947)^(1/100) = 0.00051.

Onto Lecture 42