ergodicity

Watterson GA 1975 On the number of segregating sites in genetical models without recombination. Theor Popul Biol 7:256-276.

  • by a somewhat heuristic argument, implicitly applying an ergodic theorem to interpret the results for one site to many sites, and carrying the results for a branching process model over to a model of fixed population size of the Wright type, Fisher found that the mean number of segregating sites would be as given by (1.12) above, with θ = 2
  • for independent sites with Poisson mutations, Ewens (1974) found that K2 would be exactly Poisson distributed in Wright's model, with mean θ and (in contrast to (1.10)) variance θ
  • the difference is due to correlation between sites in our models
  • Kimura (1969) had found that the mean of K2 would be θ and the variance would be (2/3)θ, for an independent sites model
  • he assumed that there was a fixed number νm of mutations per generation
  • he did not specify how these were distributed among the sites
  • he implicitly used an ergodic argument to convert results for one site to many sites
  • he also ignored the variability of nucleotide frequencies at that site, as studied later, for instance, by Maruyama (1973) and Watterson (1974)
  • the independence of sites assumption employed by Fisher (1930), Kimura (1969), Maruyama (1973), Ewens (1974), and others, seems a particularly dangerous one considering the biology of the situation, except when only mean results are required
  • sites can be correlated due to their belonging within gametes, even if they are on different chromosomes and so are unlinked
  • each gamete should inherit its sites from at most two parental gametes, rather than each site being chosen independently across the whole population
  • the same criticism then applies to an across-sites ergodic argument based on this assumption
  • there is, however, a perfectly valid ergodic argument that the stationary mean number of segregating sites, E(K2N), should equal the product of the mutation rate, νm, per generation of 2N gamete deaths, and the mean number of generations, E(T) say, that a given site with initial mutant relative frequency p = 1/2N takes to become homozygous again, due to the drift towards fixation or loss of the mutant