polygenic adaptation - senecio7の日記

Lynch M & Abegg A 2010 The rate of establishment of complex adaptations. Mol Biol Evol 27:1404-1414.

a central problem in evolutionary theory concerns the mechanisms by which adaptations requiring multiple mutations emerge in natural populations
we develop a series of expressions that clarify the scaling of the time to establishment of complex adaptations with population size, mutation rate, magnitude of the selective disadvantage of intermediate-state alleles, and the complexity of the adaptation
even in the face of deleterious intermediate steps, the time to establishment is minimized in populations with very large size
the time to establishment also scales by no more than the square of the mutation rate, regardless of the number of sites contributing to the adaptive change
the emergence of complex adaptations is only weakly constrained by the independent acquisition of mutations at the underlying sites
these results illustrate the plausibility of the relatively rapid emergence of specific complex adaptations by conventional population genetic mechanisms and provide insight into the relative incidences of various paths of allelic adaptation in organisms with different population genetic features

understanding the mechanistic origins of complex adaptations (here defined as character alterations requiring more than one novel mutation to yield a functional advantage) remains a central challenge for evolutionary biology
whereas large populations provide more individual targets for mutational origin, should the intermediate steps toward a complex adaptation be disadvantageous, the increased efficiency of selection against intermediate mutants in large populations might inhibit adaptational advance
the steady input of new mutations results in the maintenance of a small stable reservoir of intermediate alleles poised to take the next step(s) in the path toward adaptation

the rate of tunneling with a neutral intermediate has been worked out by Komarova et al. (2003) and Iwasa et al. (2004) in an application of the Moran model
after accounting for diploidy and the 2-fold reduction in the rate of drift with the Wright–Fisher model, the rate of appearance of the first double mutant destined to fixation by tunneling becomes
r_t ≅ (1 − e^−4Nu) √(uφ₂) ... (3a)
two modifications appear
1) in prior applications, the leading term in this expression has been 4Nu rather than (1 − e^−4Nu)
this has constrained the generality of tunneling theory to situations in which 4Nu < 1
because the probability that a given generational cohort can ultimately fix in a population cannot exceed 1.0
this limitation is dealt with by the leading term in equation (3a), which is well approximated by 4Nu when 4Nu ≪ 1, but asymptotically approaches one as 4Nu → ∞
2) in prior applications, the initial frequency of a mutant allele has been assumed to equal 1 / (2N)
in sufficiently large populations (2Nu > 1), more than one mutant allele appears per generation, so a more appropriate estimate of the probability of fixation of a secondary mutation is obtained by solving equation (1) with p = [1 + (2N − 1)u] / (2N)
(one mutation has definitely arisen, but a fraction u of all other members of a cohort will also have accumulated mutations at the same time)

following the logic leading to the development of equation (3a), earlier results of Komarova et al. (2003) and Iwasa et al. (2004, 2005) can then be modified to obtain an analytical approximation for the rate of tunneling
r_t ≅ 1 − e^{−4Nu²φ₂ / s₁}
φ₂ is evaluated with p = 1 / (2N)
in each generation, there are an expected 4Nu / s₁ copies of the first-step mutation segregating in the population, a fraction u of which are converted to second-step mutations, which fix with probability φ₂
as above, use of the exponential approximation accounts for the fact that no more than one tunneling event can occur per cohort