markovian

Gillespie JH 2004 Why k = 4Nus is silly. Pages 178-192 in Singh RS & Uyenoyama MK, eds. The evolution of population biology. Cambridge UP. ISBN:9780521814375

  • k = 4Nus ... (9.1)
  • Wright (1949) appears to be the first to have used this formula
  • Kimura (1971) was the first to apply it to molecular evolution
  • Equation 9.1 has been used by Kimura and Ohta (1971) to argue that the molecular clock is incompatible with molecular evolution driven by natural selection
  • Gillespie (1999, 2000) has shown that the rate of substitution (of sites in an infinite-sites, no-recombination model) is a concave rather than a linear function of the population size
  • given that one substitution has occurred, the subsequent substitution must begin as a mutation of the first substituting allele
  • Gerrish and Lenski (1998) have called this clonal interference
  • it is but one instance of many where intuition based on two-allele results (k = 4Nus comes from a two-allele model) fails when more alleles are present
  • there is another more compelling reason
  • in deriving k = 4Nus, there is an implicit assumption that s is the same for each substitution
  • the general paradigm of Darwinian evolution would have each substitution improving a species' fit to its environment
  • we would expect the long-term rate of substitution to be effectively zero and, trivially, independent of u, s and N
  • Mt = n means that in a time span of t generations there were n changes in the environment
  • the total number of substitutions in t generations may now be written as
  • St = X1 + X2 + ⋅⋅⋅ + XMt ... (9.2)
  • k = limt→0 E{X}E{Mt} / t = λ E{X} ... (9.3)
  • λ is the rate of change of the environment
  • if λ is so large that the environment changes before a substitution is completed, then Equation 9.3 no longer holds
  • I would argue that the latter comes much closer to our usual sense of adaptive evolution
  • changes in the environment lead to gene substitutions
  • it comes very close to Fisher's (1958) view of evolution in response to the "deterioration of the environment"
  • it is quite different from Wright's models based on adaptive landscapes
  • "a simple stochastic gene substitution model" (Gillespie 1982)
  • the mutational landscape model (Gillespie 1984)
  • there is recurrent mutation to a finite number of alleles that are one mutational step away from the fixed allele rather than nonrecurrent mutations to an infinity of alleles
  • the simple model shares important properties with the house-of-cards model
  • each burst of substitutions involves only a small number of alleles
  • the fitness jumps in the early substitutions are larger than those of subsequent substitutions
  • the mutational landscape model represents a step toward a more realistic mutational structure
  • each allele mutations to a finite number of other alleles
  • each of these mutant alleles in turn mutates to their own finite set of unique alleles
  • there is an infinity of alleles
  • but only a finite number are available at any time
  • we should entertain the notion that the time scale of environmental change may be much shorter than the time scale of molecular evolution
  • if so, it necessitates abandonment of the representation of St
  • our current state of knowledge does not allow us to speculate on the tie scale of environmental change experienced by a locus
  • the issue is further complicated by the fact that the relevant environment for many loci is almost certainly determined by evolving epistatic interactions