2020-05-23

population expansion

Johri P, Charlesworth B & Jensen JD 2020 Toward an evolutionarily appropriate null model: jointly inferring demography and purifying selection. Genetics 215:173-192.

previous studies have inferred ~two- to fourfold growth in the Zambian population of D. melanogaster (Ragsdale and Gutenkunst 2017; Kapopoulou et al. 2018)
we infer only 1.2-fold growth, with an ancestral N_e of 1,225,393 and current N_e of 1,357,760
as shown in Figure 6, we infer a much larger proportion of mildly deleterious mutations and a smaller proportion of highly deleterious mutations than in previous studies
f₀ = 24.7%, f₁ = 49.4%, f₂ = 3.9 %, and f₃ = 21.9%
this reflects the fact that our procedure includes synonymous sites among the total

not only does a discrete DFE provide flexibility in inference, it may also be a more realistic representation of the true DFE
gamma distributions represent a reasonably good fit to the DFE inferred from genome-wide studies
the DFE will be misinferred if the true distribution is multimodal (Kousathanas and Keightley 2013), as has been widely observed
e.g., in yeast (Bank et al. 2014a), viruses (Sanjuán 2010) and Escherichia coli (Jacquier et al. 2013)

2020-05-02

separation of timescales

Martin G & Lambert A 2015 A simple, semi-deterministic approximation to the distribution of selective sweeps in large populations. Theor Popul Biol 101:40-46.

the main tool to obtain the distribution of allelic frequencies at at given time is by numerical solutions of the diffusion equation by perturbation analysis for small N_es (Kimura, 1957), or more recently, via spectral analysis of the diffusion operator (Song and Steinrücken, 2012)

we start from the Wright-Fisher diffusion approximation of the exact stochastic process of allelic frequency under selection and genetic drift
we approximate this diffusion itself, via a well-known separation of timescales
this separation of timescales is explicitly connected to the simpler Feller diffusion process

the dynamics of p_t follow a stochastic differential equation (SDE) corresponding to the Wright-Fisher diffusion approximation
dp_t = p_t q_t (s_a q_t + s_b p_t) dt + √(p_t q_t / (2N_e)) dB_t ... (1)
s_a = s (h + (1 − h) F)
s_b = s (1 − (1 − F) h)
p₀ = k / 2N
the haploid model is characterized by s_a = s_b = s_*
conditional on fixation, p_t is in fact only 'stochastic' while p_t → 0 (early phase A: t → 0) or q_t → 0 (late phase C: t → τ), the intermediate phase B being approximately deterministic
the resulting early phase A is characterized by a simpler (linear) diffusion (Feller, 1951), akin to a branching process with independently growing types, as first noted by Haldane (1927)
the same goes for the late phase C
phase A : p_t = o(1) : dp_t ≈ s_a p_t dt + √(p_t / (2N_e)) dB_t + O(p_t²dt) ... (2)
phase B : p_t = O(1) : dp_t ≈ p_t q_t (s_a q_t + s_b p_t) dt + O(1 / √(N_e) dt)
phase C : q_t = o(1) : dp_t = −dq_t ≈ s_b q_t dt + √(q_t / (2N_e)) dB_t + O(q_t²dt)
phase A in Eq. (2) is the SDE of a supercritical Feller diffusion (Feller, 1951) with 'drift' term s_a > 0 and 'diffusion' term 1/2N_e
p_t ~ Feller(s_a, 1/2N_e)
q_t ~ Feller(−s_b, 1/2N_e)

conditional on non-extinction, a Feller diffusion starting at some p₀ > 0, when properly rescaled by its expectation E(p_t) = p₀ e^s_at, converges to some fixed distribution as time grows to infinity
more precisely, define the positive random variable lim_t→∞ ω_t = p_t / E(p_t) (called a 'terminal value')
for a trajectory destined to fixation, the distribution of this variable converges, at large times, to an exponentially distributed random variable ω_t = ω^~ ~ Exp(P₁)
P₁ is the non-extinction probability of the Feller diffusion
any density independent branching process ends up being roughly deterministic and exponentially growing at a fixed rate, if avoiding initial extinction
the long term behavior of non-extinct processes thus appears as a distribution of exponentially growing processes, all with same rates (s_a here) and variable 'starting points' (ω^~p₀ here)
applying this property to the phase A of our selective sweep model, starting from a single copy (p₀ = 1/2N), yields the following long term behavior of phase A
lim_t→∞ p_t e^−s_at = p^~₀ ~ Exp(2NP₁)
the allele will have reached its asymptotic behavior by the time it enters phase B
the trajectories entering this phase are approximately distributed as p^~₀ e^s_at, namely as if they had been deterministic from the start, but with variable initial frequency p^~₀
if an arbitrary number 1 ≤ k ≤ n of the k initial copies reach establishment, they will do so independently, yielding several independent Feller diffusion, all conditioned by fixation, during phase A
the allele then rises into phase B approximately as the sum of n processes with i.i.d. exponentially distributed initial states, yielding a gamma distribution for p^~₀
lim_t→∞ p_t e^−s_at = p^~₀ ~ Γ(n, 1/(2NP₁))

2020-05-01

fixation time

Charlesworth B 2020 How long does it take to fix a favorable mutation, and why should we care? Am Nat 195:753-771.

R. A. Fisher independently described a deterministic model of selection at an autosomal locus with discrete generations in an article that is the ultimate source of modern population genetics
he provided an exact formula for the allele frequency of a favorable allele after n generations of selection, when the fitness of the heterozygote at a biallelic locus is the geometric mean of the fitnesses of the two homozygotes
Fisher also introduced the method of branching processes for studying the probability of survival of a new favorable mutation in a very large population
for the diffusion equation approach, Fisher assumed binomial sampling from the parental generation to generate a sample of 2N independent alleles in the offspring generation
this is now commonly referred to as the Wright-Fisher model, for reasons that are somewhat obscure
Fisher proposed the model in 1922 and Sewall Wright initiated his own work on the problem only after meeting Fisher in 1924 (Provine 1986, p.239)

according to Provine (1986, p. 237), Wright had independently derived many of Haldane's results in the manuscript that became his classic article "Evolution in Mendelian Populations" (Wright 1931) but omitted this material when he learned of Haldane's work

after corresponding with Wright, Fisher (1930) corrected an error in the use of the diffusion equation in his 1922 article, which arose from his assumption that the expected change under drift was zero for his arc cosine transformation of the allele frequency

he used the allele frequency change expression for a semidominant mutation under weak selection to obtain the well-known formula for the fixation probability of a new favorable or deleterious semidominant mutation in a finite population (Fisher 1930, pp. 215-216), as well as the probability distribution of allele frequencies under irreversible mutation
Wright (1931) used a somewhat less elegant approach to obtain the same results

A₂ behaves at first like a neutral variant with initial frequency q₀ (assumed to be close to zero)
its expected frequency at the end of S₁ (including cases when it is lost from the population) is q₀
its probability of surviving this phase is Q₁
its conditional expected frequency becomes q₀ / Q₁, which can be equated to q₁
use of Haldane's (1927b) expression for the fixation probability under a Poisson distribution of offspring gives Q₁ = s
q₁ = 1 / (2Ns) in a Wright-Fisher population

the mean time to fixation of a favorable mutation (T_f) is not very sensitive to its level of dominance for a given value of the scaled selection coefficient γ = 2N_es, provided γ ≫ 1

a recessive autosomal mutation in a randomly mating population takes much longer than a semidominant mutation to spread to intermediate frequencies from a low initial frequency
a dominant mutation takes much longer to approach fixation from a high frequency
the difference between the two classes of results arises from the important role of the stochastic phases in controlling the time taken for the initial spread of a new favorable mutation and the time taken for it to become fixed once it has reached a frequency close to 1, as was first pointed out by Ewens (1968, p.62) in the context of the fixation of a dominant mutation
these phases cover the extreme points of allele frequencies, which is where the effects of dominance are most marked and where the durations of S₁ and S₂ (T₁ and T₂) are affected in nearly opposite ways by the level of dominance
this buffers the effect of the value of h on the net time to fixation, which decreases as γ becomes larger

T₁ is always somewhat less than T₂
favorable completely or partially dominant or partially dominant alleles [?] always take longer on average to fix than completely or partially recessive alleles, as was noted previously by Teshima and Przeworski (2006) and Ewing et al. (2011)
the former are more heavily affected by the duration of S₂ and the latter by the duration of S₁
this does not contradict the often-quoted "Haldane’s sieve" in favor of dominant alleles over recessive alleles in randomly mating populations
Haldane's sieve reflects the higher probabilities of fixation of favorable alleles with higher levels of dominance, not their times to fixation conditional on fixation

the fixation time itself shows considerable stochastic variation, even in large populations

the frequently made statement that the effects of drift relative to selection become insignificant when γ is ≫1 (e.g., p. 240 of Charlesworth and Charlesworth 2010) is not completely accurate when applied to fixation time

to a high level of accuracy, little sensitivity of T_f to N was detected
the only sizeable deviations from the predictions of the approximate formulas for T_f were with γ = 2,500 and N = 1,000, for which s = 1.25, corresponding to a relative fitness 2.25 of homozygotes for the favorable mutation relative to wild-type homozygote fitness

the results derived here are therefore insensitive to population size when selection is sufficiently weak (s ≈ 0.2), vindicating the use of the diffusion equations on which they rely, even when N is as small as 1,000

this raises the question of whether serious inaccuracies are introduced by considering only the deterministic phase when calculating the effects of a sweep with partial or complete dominance or recessivity

except with very high levels of dominance or recessivity in combination with random mating, γ values that are greater than 80 should guarantee that most of the time that is relevant to recombination is contributed by the deterministic phase
except for relatively small γ values, which are expected to leave relatively small signatures of sweeps (other than at very closely linked sites), the use of the deterministic phase alone should yield accurate results
under random mating, however, the expected time during the deterministic phase with h < 0.5 that is spent with q < 0.5 is greater than the time spent with q > 0.5, and vice versa with h > 0.5
partially recessive autosomal mutations or partially recessive X-linked mutations that are not male limited in their fitness effects should cause smaller overall sweep effects on variability than semidominant or partially dominant mutations despite their slightly shorter times to fixation, as was found to be the case by Teshima and Przeworski (2006), Ewing et al. (2011), and Hartfield and Bataillon (2020) for autosomal loci
sweep duration is thus not the only determinant of variability

one final question that arises from the results described here is whether fixations of deleterious variants could contribute significantly to the observed signatures of selective sweeps
as has been known since the work of Maruyama and Kimura (1974) and is illustrated in figure 4, the time course of a deleterious mutation that is destined to be fixed by drift can be very similar to that of a favorable mutation subject to the same intensity of selection
as pointed out by Gillespie (1994), under a model of constant selection on alternative variants at a site, one favored and the other disfavored by selection, and with reversible mutation between them, the equilibrium state is an equal number of substitutions in the direction of good to bad as from bad to good, just as under the standard Li-Bulmer model of selection on codon usage
the question of whether substitutions of deleterious mutations can significantly affect patterns of variability at closely linked sites needs investigation

2020-04-17

Wright-Fisher model

Waxman D & Overall ADJ 2020 Influence of dominance and drift on lethal mutations in human populations. Front Genet 11:267.

a lethal disease-causing allele needs to be described by a modified Wright-Fisher model, which deviates from the standard model, where selection coefficients are assumed small compared with 1

most of the mutations do not conform to what is expected from the balance between mutation, purifying selection and random genetic drift
the majority of the mutations observed were at frequencies that were much elevated over what was theoretically expected
current data reveal an ascertainment bias
the disease alleles were the ones identified simply by being more frequent by chance
we present a theoretical investigation of the sensitivity of the mutation-selection dynamic to small changes in fitness of the carrier genotype, in particular when slightly overdominant

standard population genetics theory generally incorporates random genetic drift via a Wright-Fisher model (and its diffusion approximation) that is derived under the assumption of weak selection

X' = Bin(2N, X + F(X)) / (2N) ... (6)
Wright-Fisher model for weak selection
Bin(n, p) denotes a binomial random number (not a distribution), and gives the random number of successes on n independent trials, each of which has probability p of success
we need to modify the above Wright-Fisher model so it incorporates strong selection

the conventional Wright-Fisher model is based on the strong assumption that all randomness arises solely in the non-selective thinning of the population to the census population size

the conventional Wright-Fisher model also assumes census and effective population sizes coincide
it is possible to incorporate the effective population size into the Wright-Fisher model (Zhao et al., 2016)

selection is treated as a deterministic process, amounting to the population being effectively infinite during the time that selection occurs within the lifecycle
for humans in modern post-industrial populations, the number of offspring produced is typically little more than that required to replace the population
the number of zygotes produced is similar in number to the number of adults (i.e., similar to the census size)
we have used an explicitly probabilistic treatment of selection
for most practical purposes there is a negligible difference between such a model, and the model where selection is treated as acting deterministically

selection cannot be directly approximated as acting at the level of alleles (which is possible when selection is weak)
the equation X' = X + F(X) applies to the frequency of the disease-causing allele in an infinite population (see Equations 4 and 5)
we cannot simply extend this equation to become an equation describing a finite population, by using the weak-selection result
the frequency of the lethal genotype obeys the stochastic equation
X' = Bin(N, 2X + 2F(X)) / (2N) ... (7)
Wright-Fisher model for a lethal genotype
F(x) is given in Equation (5)

the frequency of the lethal allele can never exceed 1 / 2

analytical insight into the Wright-Fisher model, and the phenomena occurring in a finite population, can be gained using a diffusion approximation of this model (Kimura, 1955)

V(x) = [x + F(x)] [1 − (2x + 2F(x))] / (2N) ... (18)

an important consideration with lethality, is the explicit need to treat the action of selection on genotypes, rather than on alleles
the three genotype frequencies add to unity so just two are independent
when there is also lethality of one homozygote, this allows elimination of one of the two independent genotype frequencies, with the substantial simplification that just a single frequency is required to describe the population

for lethal alleles with a low mutation rate, even very weak overdominance can result in highly inflated equilibrium frequencies

2020-03-06

soft sweep

Hartfield M & Bataillon T 2020 Selective sweeps under dominance and inbreeding. G3:1063-1075.

a key property of soft sweeps is that the beneficial variant is present on multiple genetic backgrounds as it sweeps to fixation

recessive alleles can fix more easily in selfers than in outcrossers as homozygote mutations are created more rapidly

soft sweeps from recurrent mutation or standing variation?
sweeps arising from recurrent mutation have non–zero diversity at the selected locus
a sweep from standing variation exhibits approximately zero diversity
a sweep from recurrent mutation shows intermediate-frequency variants closer to the beneficial locus, compared to sweeps from standing variation
it may be easier to differentiate between these two sweep scenarios in self-fertilizing organisms

potential applications to self-fertilizing organisms
a recent analysis found that soft-sweep signatures can be incorrectly inferred if analyzing genetic regions that flank hard sweeps, which was named the 'soft shoulder' effect
due to the reduction in recombination in selfers, these model results indicate that 'soft-shoulder' footprints can arise over long genetic distances
one remedy to this problem is to not just classify genetic regions as being subject to either a hard or soft sweep, but also as being linked to a regions subject to one of these sweeps

2020-01-20

polygenic adaptation

Sella G & Barton NH 2019 Thinking about the evolution of complex traits in the era of genome-wide association studies. Annu Rev Genom Hum Genet 20:461-493.

the genetic architecture of a trait is informative about the population genetic processes that gave rise to it
a better understanding of these processes can provide insight into practical questions, such as
why GWASs have accounted for only a modest fraction of heritable variation
the missing-heritability problem
why GWASs differ in success across traits
the design of future mapping studies
prospects for predicting phenotypes from genomes
combining information about the effects of individual loci on traits with other kinds of data—notably, changes in allele frequency across space and time—should help us learn about polygenic adaptation

Pritchard and colleagues (17, 100) have recently proposed a thought-provoking interpretation of these findings
in their omnigenic model, the value of a given trait is determined by the expression level of a few core genes in relevant tissues
the expression levels of core genes, however, can be weakly affected by the expression level of any other gene in these same tissues, and thus by any variants that affect the expression of these other genes
weak effects could arise through shared cellular machinery or by regulatory crosstalk within the RNA and protein network
even though these trans effects may be minute, their sheer number (together with the fact that, due to selection, only weak allelic effects are expected to segregate at appreciable frequencies) leads them to account, in aggregate, for most of the heritable variance in the trait
for example, active chromatin in relevant tissues, rather than specific functional annotations, seems to be the best predictor of GWAS signals
it provides an explanation for why so many loci, distributed across the genome, contribute to the heritability of a given trait

the bulk of genetic variance in quantitative, complex traits is additive
the processes that generate an individual's phenotype clearly involve highly complex and nonlinear interactions among many genes and external and internal environments
the predominance of additive variance implies that (on average) the deviation of an individual's trait value from the population mean, z, can be well approximated by a simple additive model
z = g + e = Σ_l=1^L(a_l + a'_l) + e ... 1
a_l and a'_l are the effects of the parents' alleles at site l, scaled such that the population mean contribution at each site equals 0
a_l and a'_l reflect marginal allelic effects, averaged over the distribution of genetic backgrounds and environments in the population being considered
assuming that segregating sites are biallelic, a given site with difference a in the effect of alleles and MAF x contributes 2a²x(1 − x) to genetic variance

we would like to know how well the breeding value can be approximated by polygenic scores, constructed by adding up estimated allelic effects at subsets of loci associated with a trait
the narrow-sense heritability for traits is typically substantial (~0.1–0.9) (178), suggesting that polygenic scores should, in principle, account for a substantial proportion of the phenotypic variance
a current polygenic score for height, for example, for which h² ≈ 0.8, relies on effect-size estimates for 20,000 SNPs and explains ~40% of the phenotypic variance in the self-described "white British"

the additive component can contribute the bulk of genetic variance even when dominance and epistatic interactions are included
this argument does not assume that interactions have a negligible effect
it states that even when their effect is substantial, most of it can be attributed to the marginal effects of individual alleles

many studies have reported dominance and epistatic interactions in crosses between inbred lines and between individuals from diverged populations and species
such crosses distort the allelic frequency spectrum (see 107) and bring together combinations of alleles that selection might prevent from cosegregating in the same population (akin to Dobzhansky-Muller incompatibilities in speciation)
considerable efforts to detect epistatic interactions in human GWASs have by and large come up empty-handed

contrary to the premise of the breeder's equation, covariance between fitness and a trait might not reflect selection on the trait
it could reflect a nonheritable environmental effect on both
both fitness and the trait may be strongly affected by an individual's general condition
variation in condition may be largely due to the environment
the prevalence of disruptive and stabilizing selection in meta-analyses largely reflects a combination of low statistical power and publication bias
Sanjak et al. (147) recently applied the multivariate approach, using lifetime reproductive success as a proxy for fitness in a sample size of more than 250,000 from the UK Biobank (168), orders of magnitude larger than previous studies
they found a large excess of stabilizing relative to disruptive selection

limits on the number of selected traits
consider, for example, n uncorrelated traits that are normally distributed around the optimal n-dimensional phenotype, each with variance V_P
each trait is subject to Gaussian stabilizing selection of strength V_S^{− 1}
V_S ≫ V_P
variation in all traits will reduce fitness by ~exp(−n(V_P/2V_S)) relative to the optimal phenotype
this constraint would be weaker if most selection is soft, e.g., based on competition between individuals rather than on differences in absolute fitness
or if there are negative epistatic interactions in effects on fitness

the standard deviation in relative fitness increases with the number of traits as ~exp(n(V_P/2V_S)²)
while the standard deviation in fitness is difficult to measure (but see 20), it is unlikely to exceed ~1/4 and is bound from above by offspring numbers
assuming the traditional estimated of V_P/V_S ~ 1/20 (174) would suggest an unreasonably low number of ~400 traits
although V_P/V_S is likely smaller typically, and thus the number of traits could be somewhat larger

major loci
i.e., loci that contribute substantially to genetic variance
theory suggests that the conditions in which migration-selection balance produces major loci are quite restrictive
mutations affecting the selected trait should be clustered in low-recombination regions

V_M/V_E, known as the mutational heritability, are typically ~0.0006–0.006
V_M ≅ 2U ⋅ E(a²), [...], is likely smaller than 0.1 ⋅ V_P
the estimates of V_M/V_E suggest remarkably high mutation rates of ~0.006–0.06 per gamete per generation affecting a variety of traits
assuming a typical mutation rate of ~10⁻⁸ per base pair per generation for multicellular eukaryotes, these values suggest mutational target sizes of ~0.15–1.5 Mb, echoing the evidence for high polygenicity based on the response to artificial selection and human GWASs

V_A/V_M
this ratio, also referred to as persistence time, is a measure of the number of generations required for mutation to replenish quantitative genetic variation
if most variation is effectively neutral, the persistence time is much longer, on the order of twice the effective population size, 2N_e
estimates of the persistence time are typically on the order of 100 generations (72, 115), much lower than for any species, suggesting that the bulk of quantitative genetic variation is selected against (as opposed to being neutral or under balancing selection) and lending strong support to the dominant role of mutation-selection balance in maintaining quantitative genetic variation

one of the early fault lines, the so-called Gaussian versus house of cards debate (174), centered on whether the distribution of allelic effect sizes should be modeled as Gaussian (83, 92) or rather reflects an (unknown) distribution for newly arising mutations (141, 174, 194)
one might expect an approximate Gaussian distribution if segregating variation at a locus consisted of (largely) nonrecombining haplotypes carrying many mutations affecting the trait
this would occur if the per-site mutation rate, u, far exceeds the recombination rate, r
typically u/r ~ 1 or less for all mutations
for mutations affecting a given trait, it is likely to be at least an order of magnitude smaller
e.g., in humans, only ~10% of mutations are under selection
for this reason and others, the Gaussian assumption has been largely abandoned
LD is now thought to play a minor role in mutation–selection balance
models of mutation-selection balance over the past three decades disagree primarily in their assumptions about the relationship between allelic effects on a trait and on fitness

in the Eyre-Walker model, mutations with large effect sizes and small selection coefficients are too rare to contribute substantially to genetic variance
strongly selected mutations with large effect sizes have the greatest contribution to variance
in the Caballero et al. model (first presented in 80), the distribution of effect sizes conditional on a given selection coefficient can have thicker tails
weakly selected mutations with larger effect sizes make the greatest contribution
their diverging predictions suggest that we need to understand more about the statistical relationship between selection coefficients and effect sizes to know what to expect

3.2.1. possible extensions
one would be to relax the assumption that the (effective) number of traits affected by mutations is fixed
another generalization would be to relax the assumption that all selected traits are subject to stabilizing selection
one might also envision mutational effects on a focal trait to partially reflect contributions that are not selected
such effects can be incorporated by adding an underlying neutral trait
one might consider the effects of selected traits that are not highly polygenic

the number of strongly selected mutations that entered the population was high
the expected number per generation is 2NU
their initial frequencies were low, i.e., 1/2N
the total, strongly selected variance is approximately unaffected
the two effects cancel out
the distribution of the variance among sites is profoundly affected, with many more sites segregating but at proportionally lower MAFs, and thus with lower per-site contributions to genetic variance
weakly deleterious variation, by contrast, would have largely arisen before or during the out-of-Africa bottleneck, which implies a lower input of mutations and stronger genetic drift that accelerated the loss of most mutations and boosted the frequency of those remaining
we would therefore expect fewer weakly selected segregating sites, but with greater MAFs and per-site contributions to variance
more generally, we expect the total contribution of strongly or weakly selected mutations to variance to be fairly insensitive to changes in population size (and instead depend primarily on their mutational input)
the number of segregating sites and the distribution of their contributions to variance should be markedly influenced, with strongly selected variation more affected by the more recent population sizes than weakly selected variation

when GWS associations are represented in terms of their estimated frequencies and effect sizes, they are often distributed tightly above the variance threshold
the question about missing heritability can be recast as asking where the remaining loci reside on such plots
are they mostly strongly selected loci with relatively large effects, which evade identification because their minor alleles are so rare
or are they relatively weakly selected loci with relatively small effect sizes, which evade detection because of their small contributions to variance
fitting evolutionary models to GWAS findings can help to answer these questions

Agarwala et al. (2) took a pioneering step in this direction
they used forward population genetic simulations to generate samples of genomes
they ascribed liabilities to these genomes under a range of models that vary both in their mutational target size for the disease (which determined how many of the selected sites were picked to be causal) and in the coupling between selection and effect size, assuming Eyre-Walker's model (which determined how effect sizes were ascribed to these causal sites)
they performed GWASs on their simulated data sets and compared the numbers of GWS associations with the one observed for type 2 diabetes
they were able to rule out the pleiotropic and direct selection extremes of Eyre-Walker's model
they were left with a wide range of possible genetic architectures.

Mancuso et al. (108) applied a similar approach to the study of prostate cancer in men of African ancestry
they relied on targeted sequencing at 63 loci found to affect disease risk in a larger GWAS and estimated that rare variants at these loci (with MAFs of 0.1–1%) account for ~12% of the heritability in risk (on the liability scale) compared with ~17% for common variants
the contribution of rare variants far exceeds the neutral expectation, indicating that variation affecting disease risk is subject to purifying selection
Mancuso et al. (108) then used their heritability estimate as a summary statistic to infer the coupling between selection and effect size, again assuming Eyre-Walker's model
as in the study by Agarwala et al. (2), they were left with a wide range of possible architectures

Simons et al. (157) took a step in this direction by asking whether the distribution of variances among the loci identified in GWASs for height and BMI in Europeans accords with their theoretical predictions
assuming the loci are highly pleiotropic and under moderate to strong selection (because otherwise they contribute much less to variance), the distribution of variances among them is well approximated by a closed form that depends on a single parameter, v_S
they used the observed distribution for the ~700 GWS associations for height and ~80 for BMI to estimate this parameter
the theoretical distribution provided a good fit for either trait
models with low pleiotropy did not
moderately and strongly selected mutations affecting height have a target size of ~5 Mb and account for ~50% of the heritable variance
for BMI these values are ~1 Mb and only ~15%, respectively
they do not account for the effects of historical changes in population size
simulations that incorporate changes in the population sizes of Europeans suggest that only moderately selected loci (s ~ 10⁻³) would be identified by these GWASs
the contribution to variance per segregating, strongly selected locus has been reduced by population growth after the out-of-Africa bottleneck
their estimates should be attributed only to moderately selected loci
the heritability that they do not account for could be due to loci under stronger and/or weaker selection

current GWS loci likely reflect a fairly narrow range of selection effects
further progress therefore depends on moving beyond these GWS associations

most approaches rely on strong assumptions about genetic architecture and are sensitive to varying these assumptions
earlier work assumed that the contributions of SNPs to genetic variance are normally distributed and do not depend on their frequency
E(a²|x) ∝ [x(1 − x)]⁻¹
more recent work assumed the more flexible α model
E(a²|x) ∝ [x(1 − x)]^α
estimates of α are generally negative
mutations with larger effect sizes are more strongly selected against
there is little reason to think that selection produces genetic architectures that are well approximated by the α model

4. polygenic adaptation
many selected traits are highly polygenic
the adaptive response to changing selective pressures must often involve shifts in such traits, accomplished through changes to allele frequencies at the many segregating loci that affect them
we would therefore expect polygenic adaptation in complex traits to be ubiquitous

models of polygenic adaptation
a sudden change of environment induces an instantaneous shift in the optimum of a trait under stabilizing selection
this simple scenario provides a sensible starting point for thinking about polygenic adaptation in nature

polygenic adaptation can be so rapid because it requires only tiny changes to allele frequencies at the numerous loci contributing to genetic variation

E(Δx) ≅ 1/(2V_S) D(t)ax(1 − x) − 1/(4V_S) a²x(1 − x)(1/2 − x) ... 3.
the first term on the right-hand side corresponds to directional selection
it pushes alleles with effects that are aligned with the shift to higher frequencies
its strength weakens as the distance to the new optimum, D, decreases
the second term corresponds to stabilizing selection
it acts to reduce the frequency of minor alleles regardless of the direction of their effect, and dominates when D is small
it shapes the genetic architecture at mutation–selection–drift balance, prior to the shift in optimum
when D = 0
the expected contribution to variance is then greatest for sites with intermediate and large effects, where it is approximately constant at E(2a²x(1 − x)) ≅ v_S
for such sites, on average, x(1 − x) declines roughly as 1/a²
immediately after the shift in optimum, the change in allele frequency due to directional selection is proportional to ax(1 − x) and thus is greater for alleles with intermediate effect sizes than for alleles with large effects
the contribution of an allele to phenotypic change is proportional to a²x(1 − x) and is therefore fairly insensitive to its effect sizes
once the population mean has largely caught up with the new optimum (and D is small), stabilizing selection dominates the allelic dynamics
large-effect alleles started at low frequencies and are not likely to have neared a frequency of 1/2 by this time
they are highly unlikely to further increase in frequency, let alone fix, with stabilizing selection now acting against them
alleles with intermediate effects that are aligned with the shift reach higher frequencies by this time and are therefore more likely to increase in frequency at this second stage, eventually leading to an excess fixation of intermediate-effect, aligned alleles
as a result, and counterintuitively, over longer timescales, the contribution of intermediate-effect alleles to phenotypic change supplants the contribution of alleles with large effect
eventually, the contribution of all transient frequency changes is replaced by fixations at a small subset of loci

our current understanding provides a basis for a few useful, educated guesses
polygenic adaptation likely has minimal effects on the genetic architecture of a trait
at most loci it causes only tiny changes to allele frequencies and thus only weakly perturbs the distribution of allele frequencies
polygenic adaptation arises predominantly from standing variation and primarily from variants that were segregating at relatively high MAFs, which would suggest modest effects on levels of neutral genetic variation at linked sites
identifying polygenic adaptation is likely to require pooling the evidence for changes in frequencies across many loci affecting a given trait

4.2. identifying polygenic adaptation in humans
all the methods recently proposed for identifying polygenic adaptation in humans are based on combining signals of changes in allele frequency across many loci that affect a given trait and testing whether these changes tend to affect the trait in a given direction
the reliance on subtle signals aggregated over many loci identified in GWASs renders tests extremely sensitive to systematic biases
the first set of methods relies on frequency differences of trait-increasing alleles among extant populations
the second set instead leverages genealogical footprints of past increases in the frequency of trait-increasing (or trait-decreasing) alleles in a single population
all current GWASs employ controls for population structure
it is currently not known how well these correct for biases in tests of polygenic adaptation
much of the existing evidence for polygenic adaptation is driven by subtle biases in GWAS estimates

the first approach relies on allele-frequency differences among extant populations
or, by extension, in archaic ones
the second set of methods relies on genealogical signatures of past increases in the frequency of trait-increasing (or trait-decreasing) alleles in a single population
Field et al. (45) introduced the first method in this vein
they reasoned that a recent and rapid change in allele frequency driven by selection would result in shorter terminal branches in the genealogy of the favored allele
the haplotypes flanking the beneficial alleles should carry fewer mutations that are singletons in the sample
they relied on the distribution of distances to the nearest singleton across individuals carrying each genotype in order to estimate the (log) ratio of the mean tip-branch lengths corresponding to the two alleles
they then standardized these estimates within bins of derived allele frequencies to define the singleton density score (SDS)
two recent methods take the same general approach but rely on explicit inferences of the genealogies of SNPs associated with a trait rather than summaries of tip-branch lengths
Edge & Coop (39) relied on estimated genealogical trees to approximate the time course of allele frequencies at SNPs associated with a trait and used them to approximate the time course of the polygenic score
in principle, the test should be well powered farther back in time than one based on SDS, with power increasing with sample size
the reliability of the estimated time course inevitably degrades substantially farther back in time
the number of lineages remaining in the genealogy of the sample decreases
thus so does power
in practice, the performance of this method strongly depends on the quality of the genealogical inference
Speidel et al. (166) introduced a promising and computationally efficient method for reconstructing genealogical trees in large samples, alongside a new test for polygenic adaptation
they considered the number of lineages at the time of the most recent common ancestor of the focal allele and tested whether the branching rate that led to the current frequency of the allele is significantly greater than that of the other lineages present when the allele first arose
significance levels were derived based on the symmetry of branching rates expected under neutrality and thus depend only on the topology of the inferred tree

much of the reported evidence for polygenic adaptation in height, and plausibly in other traits, was driven by subtle, systematic biases in GWASs

assuming these problems are overcome and polygenic adaptation in multiple traits and populations is identified, what is next?
one challenge will be to place adaptation events in a given trait in the context of human evolutionary history
another challenge will be to home in on targets of selection
e.g., to infer whether signals of adaptation for multiple traits in a given population reflect selection on each of the traits or rather selection on fewer traits that are genetically correlated with the others
Berg et al. (15) tested for correlations between polygenic scores in extant populations and ecological variables, controlling for the relatedness among populations

we now think that ubiquitous heritable variation in complex traits is maintained primarily by a balance between mutation and selection
we also think that polygenic adaptation via complex traits should be ubiquitous and a major mode of adaptation
we can learn about polygenic adaptation in recent human evolution by combining GWAS data with population genetic analyses

we could envision relating the processes that shape heritable variation with the cellular processes that translate this variation into phenotypic differences
we also have evolutionary models that relate mutational effects on multiple, selected complex traits with the genetic architecture of a given trait
we can ask, e.g., how cis-acting mutational effects on gene expression translate into contributions to trait heritability
doing so may help to explain why the heritability of many complex traits is widely distributed across the genome, rather than being concentrated around specific genes and pathways that are important to those traits
low-frequency, large-effect associations are more indicative of genes that directly affect a trait
common, smaller-effect ones are indicative of more general attributes of gene regulatory networks
the integration of evolutionary models and GWAS data may also turn out to be key to learning about biology from GWASs

2020-01-19

polygenic adaptation

Hayward LK & Sella G 2019 Polygenic adaptation after a sudden change in environment. bioRχiv 792952.
doi:10.1101/792952

hard sweeps were rare, at least over the past ~500,000 years of human evolution (13, 14)

for many traits, estimates of the heritability contributed by chromosomes are approximately proportional to their length (17), suggesting that the contributing variants are numerous and roughly uniformly distributed across the genome

recent studies pooled signatures of frequency changes over the hundreds to thousands of alleles that were found to be associated with an increase (or decrease) in a given trait

quantitative traits are unlikely to be subject to long-term continuous change in one direction
they are often subject to long-term stabilizing selection (3), with intermittent shifts of the optimum in different directions
the second scenario therefore assumes that a sudden change in the environment induces an instantaneous shift in the optimum of a trait under stabilizing selection

with GWASs now enabling us, at least in principle, to learn about the genetic basis of the phenotypic response, we would like to understand the allelic dynamics that underlie it

we follow previous work in considering the phenotypic and allelic responses of highly polygenic traits after a sudden change in optimal phenotype
we do so in finite populations and employ a combination of analytic and simulation approaches to characterize how the responses varies across a broad range of evolutionary parameters

we assume a Gaussian (absolute) fitness function:
W(z) = exp(− z²/2V_S) ... (2)
[absolute?]

parents are randomly chosen to reproduce with probabilities proportional to their fitness
i.e., Wright-Fisher sampling with fertility selection
[fertility?]

√(2NU)≫1
U = Lu≫1
the mutational target size, L, would have to exceed ~5 Mb
s_e = a²/V_S≪1
we assume that a substantial proportion of mutations are not effectively neutral
quantitative genetic variance is not predominantly neutral
δ² = V_S/2N
√V_A≫δ
Λ > δ
Λ < √V_S
a≪√V_S
s_d = 2Λ⋅a/V_S≪1

the closest previous work assumed an infinite population size (36, 37, 40, 43)
relaxing this assumption leads to entirely different behavior
variation in allele frequencies due to genetic drift, which is absent in infinite populations, critically affects the allelic response to selection

an allele’s contribution to phenotypic change is proportional to its contribution to phenotypic variance before the shift
alleles with moderate and large effect sizes make the greatest per site contributions to phenotypic change
alleles with moderate effect sizes experience the greatest frequency changes

the transient contributions of large effect alleles are supplanted by contributions of fixed moderate, and to a lesser extent, small effect alleles
this process takes on the order of 4N_e generations, after which the steady state architecture of genetic variation around the new optimum is restored

our finding that large effect alleles almost never sweep to fixation appears at odds with the results of previous studies of similar models
these discrepancies are largely explained by earlier papers considering settings that violate our assumptions, notably about evolutionary parameter ranges
some studies assume that large effect alleles segregate at high frequencies before the shift in optimum (e.g., (65)), which is presumably uncommon in natural populations and in any case, violates our assumption that the trait is at steady state before the shift
Thornton (38) observes sweeps in cases in which the trait is not highly polygenic (violating our assumption that √(2NU)≫1)
Chevin and Hospital (66) observe sweeps in cases in which a single newly arising mutation of large effect contributes substantially to genetic variance, which violates our assumptions that genetic variation is highly polygenic and is not predominantly effectively neutral
quantitative genetic variation is not predominantly neutral
Stetter et al. (39) considered a huge shift in the optimal trait value (e.g., of ~90 phenotypic standard deviations), resulting in a massive drop in fitness (violating our assumption that Λ < √V_S)
little is known about the magnitude of shifts in optimal trait values over the time scales of large effect, beneficial fixations
the response to such larger shifts is not covered by our analysis and clearly warrants further study

pleiotropy is therefore likely to affect which alleles contribute to phenotypic change at the different phases of polygenic adaptation

after a shift in the optimal trait value, the number of fixations of alleles whose effects are aligned to the shift are nearly equal to the number of alleles that are opposed (Fig. 6)
the alleles that fix are a largely random draw from the vastly greater number of alleles that affect the trait
in this plausible scenario, it becomes meaningless to say that any given fixation was adaptive, and arguably uninteresting to focus on the particular subset of alleles that happened to reach fixation
identifying the traits that experienced adaptive changes promises to provide important insights