Fixation indices

Introduction

Inbreeding

  • “Consanguineous mating”
  • Mating between relatives
  • Positive assortative mating
  • Extreme case: self-fertilization (e.g. Arabidopsis thaliana)

Self fertilization: heterozygosity

Self-fertilization & allele frequencies

  • Recall:
    • \(p = X + \frac{Y}{2}\)
  • Generation 1
    • \(X=0\), \(Y=1\)
    • \(p = 0 + \frac{1}{2} = 0.5\)
  • Generation 2
    • \(X=0.25\), \(Y=0.5\)
    • \(p = 0.25 + \frac{0.5}{2} = 0.5\)

Self-fertilization: summary

Fixation index

  • Measures departure from Hardy-Weinberg frequencies
  • Range -1 to +1
  • +ve values, excess homozygotes
  • -ve values, excess heterozygotes
  • \(F = \frac{H_E - H_O}{H_E}\)
  • \(H_E = 1 - \sum p_i^2\)
  • \(H_O = \sum Y_i\)

Subdivided populations

Sources of departure from \(H_E\)

  • Non-random mating within sub-populations (e.g. inbreeding)
  • Wahlund effect: Allele frequency differences between sub‑populations

Fixation indices

  • Introduced by Sewall Wright
  • Formulated for a dialleleic locus
  • Hierarchical
  • Can be applied to hierarchies of any depth
  • Most common: 3 levels
    • Total population
    • Sub-populations
    • Individuals in sub‑populations

Fixation indices

  • \(F_{IS}\): Effect of non-random mating within subpopulations. Correlation of alleles in individuals, relative to sub-populations
  • \(F_{ST}\): Magnitude of the Wahlund effect. Correlation of alleles within sub‑populations, relative to total population
  • \(F_{IT}\): Combined effect of non-random mating and Wahlund effect. Correlation between alleles within individuals, relative to total population

Three measures of heterozygosity

\(H_I = \frac{\sum_{i=1}^n H_{O_i}}{n}\)

Average observed heterozygosity, frequency of heterozygous individuals

\(H_S = \frac{\sum_{i=1}^n 2p_iq_i}{n}\)

Average expected heterozygosity, among subpopulations

\(H_T = 2\bar{p}\bar{q}\)

\(H_T = 2 \left(\frac{\sum_{i=1}^n p_i}{n}\right) \left(\frac{\sum_{i=1}^n q_i}{n}\right)\)

Total expected heterozygosity

Defining fixation indices

  • \(F_{IS} = \frac{H_S - H_I}{H_S}\)
  • \(F_{ST} = \frac{H_T - H_S}{H_T}\)
  • \(F_{IT} = \frac{H_T - H_I}{H_T}\)

\[1 - F_{IT} = \left(1-F_{ST}\right)\left(1-F_{IS}\right)\]

Numerical example

\(p\) \(2pq\) \(H_O\)
Population 1 0.3 0.42 0.418
Population 2 0.5 0.50 0.480
Population 3 0.8 0.32 0.336

Numerical example: \(H_I\)

\(p\) \(2pq\) \(H_O\)
Population 1 0.3 0.42 0.418
Population 2 0.5 0.50 0.480
Population 3 0.8 0.32 0.336

\[H_I = \frac{0.418 + 0.480 + 0.336}{3} = 0.4113\]

Numerical example: \(H_S\)

\(p\) \(2pq\) \(H_O\)
Population 1 0.3 0.42 0.418
Population 2 0.5 0.50 0.480
Population 3 0.8 0.32 0.336

\[H_S = \frac{0.42 + 0.50 + 0.32}{3} = 0.4133\]

Numerical example: \(H_T\)

\(p\) \(2pq\) \(H_O\)
Population 1 0.3 0.42 0.418
Population 2 0.5 0.50 0.480
Population 3 0.8 0.32 0.336

\[\bar{p} = \frac{0.3 + 0.5 + 0.8}{3} = 0.5333\]

\[\bar{q} = 1 - \bar{p} = 0.4667\]

\[H_T = 2 \times 0.5333 \times 0.4667 = 0.4978\]

Numerical example: Fixation indices

\[F_{IS} = \frac{0.4133 - 0.4113}{0.4133} = 0.0048\]

\[F_{ST} = \frac{0.4978 - 0.4133}{0.4978} = 0.1697\]

\[F_{IT} = \frac{0.4978 - 0.4113}{0.4978} = 0.1738\]

\(F_{ST}\)

  • The most commonly-computed measure in population genetics
  • Thousands of papers
  • Measure of population subdivision / genetic divergence
  • Range:
    • 0 = Identical allele frequencies in all sub-populations
    • 1 = Sub-populations fixed for different alleles

Estimating \(F_{ST}\)

  • Need estimators of \(F_{ST}\)
    • Sample data
    • Multiple alleles, multiple loci
  • Gene diversity
    • Estimators of \(H\)
    • \(G_{ST}\): Nei & colleagues
  • Analysis of variance
    • \(\theta\): Cockerham, Weir
    • \(\phi_{ST}\): Excoffier & colleagues

\(F_{ST}\) and the coalescent

\[ F_{ST}=\frac{\bar{t}-\bar{t_0}}{\bar{t}}\]

  • \(\bar{t}\): Average coalescence time for a pair of genes in the total sample
  • For finite island model: \(\bar{t}=\left(\frac{1}{d}\bar{t_0}\right) + \left(\frac{d-1}{d}\bar{t_1}\right)\)
  • \(\bar{t_0}\): Average coalescence time for a pair of genes sampled from the same deme
  • \(\bar{t_1}\): Average coalescence time for a pair of genes sampled from different demes
  • \(d\): number of demes (subpopulations)

Reading

Textbook: Pages 28 - 32; 118 - 130

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