2019-11-01

hemiplasy

Lee KM & Coop G 2019 Population genomics perspectives on convergent adaptation. Phil Trans R Soc B 374:20180236.

the sharing of traits or alleles incongruent with the population or species tree owing to ancestral variation (i.e. ILS shown in figure 1a) has been termed hemiplasy

the key question is whether selection has independently increased the allele to fixation multiple times
taxa sharing the same substitution owing to ILS or gene flow means there has not been independence at the level of the mutational change
the allele frequency change at the locus can still be independent, that is convergent, across populations
even if an allele was shared owing to ILS, selection may have repeatedly driven up the allele in each population separately

regardless of the precise terminology we use, it is important to keep clear statements about the level of independence and the role selection has played in these changes
we introduce a conceptual framework to quantify the degree to which allele frequency shifts have been independent

we should be much more impressed by selection that repeatedly moves an allele from 10⁻⁴ to 50% than 50% to 100%, despite being comparable frequency changes and taking roughly the same time under an additive model

we should be much more impressed by repeated adaptation from very rare standing variation than common variation

we cannot simply sum the work done by selection (i.e. selective deaths) across multiple loci, unless the loci contribute multiplicatively to fitness
this argument forms the basis of the rejection of Haldane's substitution load argument, and Kimura's gain of information extension, as a limit on the rate of evolution
combining information over loci additively, that is assuming multiplicative epistasis, seems a fine first approximation
this limitation should be borne in mind

convergent adaptation, at the genetic or phenotypic level, can potentially be conceptualized in this way as the independent gain of information owing to selection

dissecting whether the same or different mutations gave rise to the adaptive alleles will help us gain insights into questions related to mutational target size or epistatic constraints
these questions can be somewhat orthogonal to more basic questions about adaptation
human populations have repeatedly adapted to different environments through changes in skin pigmentation
this convergent adaptation on the level of the phenotype comes from a mixture of selection on old standing variation, both derived and ancestral variants, and recent mutations

in information theory, the negative log probability of an outcome under a particular model is called the surprise (or the self information)

2019-09-05

population expansion

Ragsdale AP & Gravel S 2019 Models of archaic admixture and recent history from two-locus statistics. PLoS Genet 15:e1008204.

human expansion models underestimate LD between low frequency variants
the demographic model for human out-of-Africa (OOA) expansion proposed and inferred by Gutenkunst et al. [2] has been widely used for subsequent simulation studies
parameter estimates have been refined as more data became available [3, 35, 38]
we generalized the Gutenkunst model with a number of additional parameters accounting for recent events, including size changes in the YRI population, recent admixture between populations, and substructure within each continental population
none of these modifications provided satisfactory fit to the data and some did not converge to biologically realistic parameters

Fig 3
(A) we fit the 13-parameter Gutenkunst et al. model to statistics in the two-locus, multi-population Hill-Robertson system
most statistics were accurately predicted by this model, including
(B) the decays of E[D²] in each population
(C) the decay of the covariance of D between populations
(E) the joint heterozygosity E[π₂(i)]
(D) E[D_i(1 − 2p_i)(1 − q_i)] was fit poorly by this model
we were unable to find a three-population model that recovered these observed statistics, including with additional periods of growth, recent admixture between human populations, or substructure within modern populations

Table 1
we fit the commonly used 13-parameter model to the multi-population Hill-Robertson statistics
the best fit parameters shown here were fit to the set of statistics without the E[Dz] terms
inclusion of those terms led to runaway parameter behavior in the optimization
this is often a sign of model mis-specification
the same 13-parameter model is augmented by the inclusion of two deeply diverged branches, putatively Neanderthal and an unknown lineage within Africa
these branches split from the branch leading to modern humans roughly 460 – 650 kya, and contributed migrants until quite recently (~ 19 kya)
times reported here assume a generation time of 29 years and are calibrated by the recombination (rather than mutation) rate

Fig 4
inferred OOA model with archaic admixture
(A) we fit a model for out-of-Africa expansion related to the standard model in Fig 3A
we also include two branches with deep split from the ancestral population to modern humans
(B-E) this model fits the data much better than the model without archaic admixture, and especially for the Dz statistics (D)

beyond the classical recursions for E[D] and E[D²] [12, 14], two-locus statistics are difficult to compute for non-equilibrium, multi-population demographic models
Sved [53] proposed an IBD based recursion to compute E[r²] across subdivided populations
its accuracy and interpretation remain debated [12]

2019-05-31

canalization

Hallgrimsson B, Green RM, Katz DC, Fish JL, Bernier FP, Roseman CC, Young NM, Cheverud JM & Marcucio RS 2019 The developmental-genetics of canalization. Sem Cell Dev Biol 88:67-79.

canalization is a potentially significant cause of missing heritability

canalization, or the tendency to buffer variation, is about the modulation of phenotypic variance due to factors other than the genetic or environmental variance, per se
phenotypic robustness is a more general concept
canalization refers to the minimization of variation among individuals
developmental stability is the tendency to minimize variation among replicated structures within individuals
developmental stability is most often measured via random, normally distributed departures from symmetry where symmetrical development is expected, or fluctuating asymmetry
canalization and developmental stability are partially overlapping phenomena
it is not known, however, to what extent measures of canalization and developmental stability capture the same biological phenomena

canalization was motivated by developmental biology
phenotypic plasticity has a more naturalistic or population biology origin
coined by Woltereck [37], the concept of a reaction norm was fleshed out by Schmalhausen [38] whose primary interest was in the central role of stabilizing selection in evolution
independently of Waddington, Schmalhausen proposed a concept very similar to canalization which, in the English translation, he termed autonomization
for him, reaction norms, shaped by stabilizing selection, are adaptive
autonomization is not just any minimization of any variation
it is specifically the minimization of non-adaptive variation around the reaction norm
this contrasts significantly with current thinking on plasticity
plasticity is not necessarily adaptive

canalization and phenotypic plasticity are merely abstractions from patterns in data
they are not processes or mechanisms

the problem with viewing canalization as simply the inverse of plasticity is that plasticity is a much more general phenomenon
Wagner et al.'s [8] definition of canalization avoids this confusion
they define canalization as the suppression of phenotypic variation of either genetic or environmental origin
this makes canalization a dispositional concept, referring to a tendency or potential
canalization is the tendency to suppress variation
canalization is not a component of an observed phenotypic variance
it is a component of variability, or the tendency to vary
defined in this way, canalization is distinct from the more general concept of phenotypic plasticity
this definition also allows for the possibility of changes in among-individual variance along the norm of reaction
such changes in among-individual variance can result from a nonlinear norm of reaction curve
the shape of such norm of reaction curves can vary among genoytpes
they might also result from other factors such as destabilizing effects of environmental stress or genetic perturbations

these early studies showed that the tendency for increased among-individual variance in mutants responded to selection, suggesting a heritable basis for canalization

all genetic canalization effects are attributable to epistasis in a quantitative genetic sense
yet large, population-level quantitative genetic effects are no assurance that underlying biological interactions between genes and/or gene products are understood or can be identified in individuals
to say epistasis explains genetic canalization in a mechanistic sense is therefore to commit Roff's sin of confounding a quantitative model with a mechanistic explanation
at the heart of this issue is the fact that quantitative genetics and developmental biology are concerned with different kinds of questions and have different definitions of ostensibly the same phenomena
quantitative genetics is concerned with partitioning phenotypic outcomes into cumulative, statistically-defined phenomena
developmental biology is concerned with physical mechanisms such as specific molecular, cellular or tissue-level interactions
the genetics of a population of clones is singularly uninteresting
its developmental biology may be fascinating
quantitative genetics defines most phenomena at the population level and is fundamentally concerned with variation
the mechanisms of interest to developmental biology occur in individuals and, at least under the current paradigm, variation is more often a nuisance than a direct object of study
in genetics, epistasis is a statistical effect
in developmental biology it is generally viewed as a mechanistic interaction between gene products
the two versions of epistasis are linked conceptually but are not identical
canalization is a property of individuals that is almost always measurable only at a population level
in genetic terms, a genetic factor increases canalization if, all else being equal, it reduces the phenotypic variance
for genetic canalization, the influence of a gene on the phenotypic effect of other genes is, by definition a case of epistasis
genetic variation in such effects could be described as differential epistasis
does this mean that epistasis fully explains genetic canalization?
not necessarily
gene interaction effects imply nonlinear effects on phenotype
this is a statistical model of variation and not a mechanistic description of actual physical interactions
nonlinearities in development arise in myriad ways
depending on one's perspective (genetic or developmental), epistasis can be seen as either a cause or a consequence of canalization

for genes with redundant functions, null mutations might, therefore, be expected to produce a viable but more variable phenotype
conversely, redundancy in gene function contributes to robustness to mutation in those genes

the mapping of gene expression to phenotype is highly nonlinear
loss of over 50% of the wildtype expression level is associated with minimal changes in phenotype
beyond this point, the phenotypic effects are large and increasingly severe
we extended variation in Fgf8 expression well beyond the range expected in natural populations
this begs the question of how such nonlinearities evolve
the local flattening of the gene-expression to phenotype around the wildtype could evolve by stabilizing selection acting on gene-interaction effects
the question of how stabilizing or directional selection influence variational properties such as canalization has been an area of significant debate

diversity, often originating in different disciplines, has also created a situation reminiscent of the parable of the blind men and the elephant
each tends to be concerned with a particular aspect of the problem, making it difficult to contextualize this progress in terms of a more general understanding of the mechanisms of canalization in development

2019-05-07

polygenic adaptation

Thompson KA, Osmond MM & Schluter D 2019 Parallel genetic evolution and speciation from standing variation. Evol Lett 3:129-141.

adaptation from standing variation would reduce the evolution of reproductive isolation under parallel selection
parental populations would fix more of the same alleles and therefore evolve fewer incompatibilities

our simulations consider pairs of populations and multivariate phenotypes determined by multiple additive loci

N haploid parents are then randomly sampled with replacement from a multinomial distribution with probabilities proportional to their fitness, W
parents then randomly mate and produce two haploid offspring per pair, with free recombination between all loci
we assume an effectively infinite number of loci such that all mutations arise at a previously unmutated locus
mutational effects are drawn from a multivariate normal distribution ("continuum-of-alleles" sensu Kimura [1965]), with a mean of 0 and an SD of α in all m traits and no correlations among traits
i.e., universal pleiotropy

our general conclusions hold if the ancestor is under much stronger selection (σ_anc = 1) that puts it into the multivariate "House-of-Cards" regime

a parental population was established by first randomly choosing n polymorphic loci in the ancestor
each parental individual received the mutant (i.e., "derived") allele at each of these n loci with a probability equal to the allele's frequency in the ancestor
this admittedly artificial sampling procedure allowed us more control over the amount of standing genetic variation across simulations with different parameter values
further control was achieved by making the second parental population initially identical to the first
each possessed the exact same collection of genotypes
there were therefore no founder effects
populations adapted from only new (i.e., de novo) mutation when n = 0

an unavoidable and important effect of standing variation is that it quickens adaptation because populations do not have to wait for beneficial alleles to arise
reproductive isolation evolves rapidly during the initial stages of adaptation
after populations reach their respective phenotypic optima, genetic divergence accumulates slowly at a rate proportional to the mutation rate
our results reflect quasi-equilibrium conditions rather than transient states and are unaffected by standing variation's influence on the speed of adaptation

phenotypic variance in parental populations (i.e., before hybridization) is near zero and does not differ between populations founded with versus without standing variation
such low variance is expected because our simulations have fixed optima, frequency-independent selection, no migration, and parameter values corresponding to strong selection and relatively weak mutation

we determined the fitness (eq. (1)) of each hybrid in both parental environments and recorded its fitness as the larger of the two values

genetic parallelism rarely decreases to zero even under completely divergent selection (θ = 180°), indicating that populations fix some deleterious alleles

2019-03-25

polygenic adaptation

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

2019-03-24

polygenic adaptation

Stetter MG, Thornton K & Ross-Ibarra J 2018 Genetic architecture and selective sweeps after polygenic adaptation to distant trait optima. PLoS Genet 14:e1007794.

during the stationary phase before the shift and after reaching the new optimum we followed a Gaussian fitness function appropriate for a trait under stabilizing selection
during the optimum shift, however, such a model would be problematic, as only a few individuals in the upper tail of the fitness distribution would have extremely high relative fitness, inducing a strong population bottleneck
instead, we applied a model of truncation selection, first calculating fitness under the Gaussian fitness function but then assigning a fitness of 1 to the top half of the population and 0 to the bottom half
such a model is reasonable for sudden shifts in trait optima that do not lead to the extinction of a population, but where higher trait values are unambiguously advantageous and the maximum population size is limited
in natural populations these factors can be observed when sudden changes in the environment favor a specific phenotype for invasive species

we modeled 20 QTL resembling 50kb regions, each with a 4 kb "genic" region centered in a 46 kb "intergenic" region
fitness
w = exp[− (z − z_opt)² / (2V_S)] ... (1)
this standard model for traits under stabilizing selection is well suited for populations at equilibrium
under strong directional selection, however, this model greatly amplifies fitness differences among individuals in the tails of the phenotypic distribution
during the adaptive phase of the simulation, we calculated individual fitness following Eq 1, but then apply truncation selection by assigning a fitness of 1 to the top 50% of the distribution of w and 0 for the remaining 50%
this model allowed for truncation selection on z, while the population was distant from the new optimum, but allows for selection against phenotypes that surpass the new optimum during the final stages of adaptation
we stopped truncation selection once the population mean reached the new optimum
initial genetic variance
the genetic variance at equilibrium can be approximated by the house of cards (HoC) model
E[V_G] = 4μV_S ... (2)
and the stochastic HoC approximation
E[V_G]_SHC = 4μV_S / (1 + V_S / (Nσ_m²)) ... (3)
background
computational limitations do not allow simulation of an entire eukaryotic genomes
we added a heritable background (G_B) to our simulations to account for the adaptive potential of the rest of the genome
sweeps
we defined as a sweep any mutation that fixed faster than 99% of neutral alleles

2019-03-23

polygenic score

Barton N, Hermisson J & Nordborg M 2019 Why structure matters. eLife 8:e45380.

the first GWAS for height found a small number of SNPs that jointly explained only a tiny fraction of the variation
this was in contrast with the high heritability seen in twin studies
it was dubbed ‘the missing heritability problem’
it was suggested that the problem was simply due to a lack of statistical power to detect polymorphisms of small effect
most of the variation remains ‘unmappable’
sample sizes on the order of a million are still not large enough
one way in which the unmappable component of genetic variation can be included in a statistical measure is via so-called polygenic scores
these scores sum the estimated contributions to the trait across many SNPs, including those whose effects, on their own, are not statistically significant
polygenic scores thus represent a shift from the goal of identifying major genes to predicting phenotype from genotype

when a GWAS is carried out to identify major genes, it is relatively simple to avoid false positives by eliminating associations outside major loci
if the goal is to make predictions, or to understand differences among populations (such as the latitudinal cline in height), we need accurate and unbiased estimates for all SNPs
accomplishing this is extremely challenging
it is also difficult to know whether one has succeeded
one possibility is to compare the population estimates with estimates taken from sibling data, which should be relatively unbiased by environmental differences

Berg et al. and Sohail et al. independently found that evidence for selection vanishes – along with evidence for a genetic cline in height across Europe
the previously published results were due to the cumulative effects of slight biases in the effect-size estimates in the GIANT data
they also found evidence for confounding in the sibling data used as a control by Robinson et al. and Field et al.
we still do not know whether genetics and selection are responsible for the pattern of height differences seen across Europe
there is no perfect way to control for complex population structure and environmental heterogeneity
biases at individual loci may be tiny
they become highly significant when summed across thousands of loci – as is done in polygenic scores
standard methods to control for these biases, such as principal component analysis, may work well in simulations but are often insufficient when confronted with real data
even the data in the UK Biobank seems to contain significant structure

quantitative genetics has proved highly successful in plant and animal breeding
this success has been based on large pedigrees, well-controlled environments, and short-term prediction
when these methods have been applied to natural populations, even the most basic predictions fail, in large part due to poorly understood environmental factors
natural populations are never homogeneous
it is therefore misleading to imply there is a qualitative difference between ‘within-population’ and ‘between-population’ comparisons