polygenic adaptation - senecio7の日記

Höllinger I, Pennings PS & Hermisson J 2019 Polygenic adaptation: from sweeps to subtle frequency shifts. PLoS Genet 15:e1008035.

population genetics views adaptation as a sequence of selective sweeps at single loci underlying the trait
quantitative genetics posits a collective response, where phenotypic adaptation results from subtle allele frequency shifts at many loci
a synthesis of these views is largely missing
we study the architecture of adaptation of a binary polygenic trait (such as resistance) with negative epistasis among the loci of its basis
our key analytical result is an expression for the joint distribution of mutant alleles at the end of the adaptive phase

for shifts, alleles need to be able to hamper the rise of alleles at other loci via negative epistasis
diminishing returns are a consequence of partial or complete redundancy of genetic effects across loci or gene pathways
adaptive phenotypes (such as [...]) can often be produced in many alternative ways
redundancy is a common characteristic of beneficial mutations

we dissect the adaptive process into two phases
the early stochastic phase descries the establishment of all mutants that contribute to the adaptive response under the influence of mutation and drift
loci can be treated as independent during this phase to derive a joint distribution for ratios of allele frequencies at different loci, Eq (5)
during the second, deterministic phase, epistasis and linkage become noticeable
mutation and drift can be ignored
allele frequency changes during this phase can be descried as a density transformation of the joint distribution
for the simple model with fully redundant loci, and assuming either LE or complete linkage, this transformation can be worked out explicitly
our main result Eq (8) can be understood as a multi-locus extension of Wright's formula
for a neutral locus with multiple alleles, Wright's distribution is a Dirichlet distribution, which is reproduced in our model for the case of complete linkage
for the opposite case of linkage equilibrium, we obtain a family of inverted Dirichlet distribution

the quantitative genetic "small shifts" view of adaptation does not talk about a stationary distribution
it does not imply that alleles will never fix over much longer time scales
the transient nature of our result means that it reflects hte effects of genetic drift only during the early phase of adaptation
our result ignores drift after phenotypic adaptation has been accomplished—which is also a reason why it can be derived at all

the qualitative pattern of polygenic adaptation is predicted by a single compound parameter: the background mutation rate Θ_bg
i.e., the population mutation rate for the background of a focal locus within the trait basis

the role of Θ_bg for polygenic adaptation is essentially parallel to the one of Θ_l for soft sweeps
the mathematical methods to analyze both cases are different
the polygenic scenario does not lend itself to a coalescent approach

alternative approaches to polygenic adaptation
the theme of "competition of a single locus with its background" relates to previous findings by Chevin and Hospital (2008) [26] in one of the first studies to address polygenic footprints
the background is modeled as a normal distribution with a mean that can respond to selection, but with constant variance
a drift-related parameter, such as Θ_bg, has no place in such a framework
a sweep at the focal locus is prohibited under two conditions
first, the background variation (generated by recurrent mutation in our model, constant in [26]) must be large
second, the fitness function must exhibit strong negative epistasis that allows for alternative ways to reach the trait optimum—and thus produces redundancy (due to Gaussian stabilizing selection in [26])
finally, while the adaptive trajectory depends on the shape of the fitness function, Chevin and Hospital note that it does not depend on the strength of selection on the trait, as also found for our model

de Vladar and Barton [42] and Jain and Stephan [31] [...] study an additive quantitative trait under stabilizing selection with binary loci
these models allow for different locus effects, but ignore genetic drift
before the environmental change, all allele frequencies are assumed to be in mutation-selection balance
sweeps are prevented in [31] if most loci have a small effect and are therefore under weak selection prior to the environmental change
this contrasts to our model, where the predicted architecture of adaptation is independent of the selection strength
in our model, weak selection does not imply shifts
this difference can at least partially be explained by the neglect of drift effects on the starting allele frequencies in the deterministic models
in the absence of drift, loci under weak selection start out from frequency x₀ = 0.5 [42]
in finite populations, however, almost all of these alleles start from very low (or very high) frequency
many alleles at intermediate frequencies at competing background loci are expected only if Θ_bg≫1, in accordance with our criterion for shifts
we have analyzed our model for the case of starting allele frequencies set to the deterministic values of mutation-selection balance, μ/s_d
we observe adaptation due to small frequency shifts in a much larger parameter range

adaptation by sweeps in a polygenic model requires a mechanism to create heterogeneity among loci

both drift and unequal locus effects are included in the simulation studies by Pavlidis et al (2012) [28] and Wollstein and Stephan (2014) [29]
due to differences in concepts and definitions there are few comparable results
they study long-term adaptation
they simulate N_e generations
sweeps are defined as fixation of the mutant allele at a focal locus
frequency shifts correspond to long-term stable polymorphic equilibria [29]
with this definition, a shift scenario is no longer a transient pattern, but depends entirely on the existence (and range of attraction) of polymorphic equilibria

the initial stochastic phase is relatively insensitive to interactions via epistasis or linkage
as described above, the key qualitative results to distinguish broad categories of adaptive scenarios are due to the initial stochastic phase
this holds true, in particular, for the role of the background mutation rate Θ_bg
we therefore expect that these results generalize beyond our basic model

N_e is the short-term effective population size [...] during the stochastic phase of adaptation
this short-term size is unaffected by demographic events, such as bottlenecks, prior to adaptation
it is therefore often larger than the long-term effective size that is estimated from nucleotide diversity