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