13: The Effects of Linked Selection

Soft Sweeps from multiple mutations and standing variation.

In our sweep model above, we assumed that selection favoured a beneficial allele from the moment it entered the population as a single copy mutation (left panel, Figure \(\PageIndex{12}\)). However, when a novel selection pressure switches on, multiple mutations at the same gene may start to sweep, such that no one of these alleles sweeps to fixation (middle panel, Figure \(\PageIndex{12}\)). These sweeps involving multiple mutations significantly soften the impact of selection on genomic diversity, and so are called ’soft sweeps’.

Another way that the impact of a sweep can be softened is if our allele was segregating in the population for some time before it became beneficial. That additional time means that our allele can have recombined onto various haplotype backgrounds, such that when selection pressures switch, the selected allele sweeps up in frequency on multiple different haplotypes (right panel, Figure \(\PageIndex{12}\)). Detecting and differentiating these different types of sweeps is an active area of empirical research and theory in population genomics (see for an overview of developments in this area).

A simple recurrent model of selective sweeps

To explain how a constant influx of sweeps could impact levels of diversity, here we will develop a model of recurrent selective sweeps.

Imagine we sample a a pair of neutral alleles at a locus a genetic distance \(c\) away from a locus where sweeps are initiated within the population at some very low rate \(\nu\) per generation. The waiting time between sweeps at our locus is exponentially distributed \(\sim Exp(\nu)\) (see math Appendix \ref{eqn:exp_rv_def}). Each sweep rapidly transits through the population in \(\tau\) generations, such that each sweep is finished long before the next sweep (\(\tau \ll \frac{1}{\nu}\)).

As before, the chance that our neutral lineage fails to recombine off the sweep is \(p_{NR}\), such that the probability that our pair of lineages are forced to coalesce by a sweep is \(e^{-c \tau}\). Our lineages therefore have a very low probability

\[\nu e^{-c \tau}\]

of being forced to coalesce by a sweep per generation. If our lineages do not coalesce due to a sweep, they coalesce at a neutral rate of \(\frac{1}{2N}\) per generation. Thus the average waiting time till a coalescent event between our neutral pair of lineages due to either a sweep or a neutral coalescent event is

\[\mathbb{E}(T_2) = \frac{1}{\nu e^{-c \tau} + \frac{1}{2N}}\]

Now imagine that the sweeps don’t occur at a fixed location with respect to our locus of interest, but now occur uniformly at random across our genome. The sweeps are initiated at a very low rate of \(\nu_{BP}\) per basepair per generation. The rate of coalescence due to sweeps at a locus \(\ell\) basepairs away from our neutral loci is \(2\nu_{BP} e^{-c_{BP} \ell \tau}\), where the factor of two comes from the fact that bases can be \(\ell\) basepairs away on the left or right. If our neutral locus is in the middle of a chromosome that stretches \(L\) basepairs in either direction, the total rate of sweeps per generation that could force our pair of lineages to coalesce is

\[2\int_0^{L} \nu_{BP} e^{-c_{BP} \ell \tau} d \ell = \frac{2\nu_{BP}}{c_{BP} \tau} \left(1-e^{-c_{BP} \tau L} \right)\]

so that if \(L\) is very large (\(c_{BP} \tau L \gg 1\)), the rate of coalescence per generation due to sweeps is \(\frac{2\nu_{BP}}{c_{BP} \tau}\). The total rate of coalescence for a pair of lineages per generation is then

\[\frac{2\nu_{BP}}{c_{BP} \tau}+\frac{1}{2N}\]

So our average time untill a pair of lineages coalesce is

\[\mathbb{E}(T_2) = \frac{1}{\frac{2\nu_{BP}}{c_{BP} \tau}+\frac{1}{2N}} = \frac{c_{BP}2N}{\frac{4N\nu_{BP}}{ \tau}+c_{BP}}\]

such that our expected pairwise diversity (\(\pi=2\mu\mathbb{E}(T_2)\)) in a region with recombination rate \(r_{BP}\) that experiences sweeps at rate \(\nu_{BP}\) is

\[\mathbb{E}(\pi) = \pi_0 \frac{c_{BP}}{\frac{4N\nu_{BP}}{ \tau}+c_{BP}} \label{eqn:pi_GW_HH}\]

where \(\pi_0\) is our expected diversity without any selective sweeps, (\(pi_0=\theta=4N\mu\)). The expected diversity increases with \(c_{BP}\), as higher recombination rates decrease the likelihood a neutral allele hitchhikes along with a sweep and is thus forced to coalesce by the sweep. Expected diversity decreases with \(\nu_{BP}\), as a greater density of functional sites experiencing sweeps increases the chance of being linked to a nearby sweep. As we move to high \(c_{BP}\), assuming that \(\nu_{BP}\) doesn’t increase with \(c_{BP}\), our level of diversity should plateau to \(\theta\), the level of genetic diversity of a neutral site completely unlinked to any selected loci. If we assume that our genome experiences a constant rate of sweeps of a given strength, i.e. that \(\frac{4N\nu_{BP}}{ \tau}\) is a constant, we can fit the variation in \(\pi\) across regions that vary in their recombination rate (\(c_{BP}\)) to estimate a population’s rate of recurrent sweeps per basepair. An example of fitting this curve to data from Drosophila melanogaster is shown in Figure \(\PageIndex{13}\); see for an early example of fitting a similar recurrent hitchhiking model to such data. The parameter giving us this best-fitting curve is \(\frac{4N\nu_{BP}}{ \tau} \approx 7 \times 10^{-9}\). With an effective population size of a million and assuming that the sweeps take a thousand generations to reach fixation, we find this implies \(\nu_{BP} \approx 10^{-12}\). Thus, a really low rate of moderately strong sweeps, roughly one every megabase every million generations, is all we need to explain the profound dip in diversity seen in regions of the genome with low recombination. However, sweeps from positively selected alleles are not the only cause of genome-wide signals of linked selection. Selection against deleterious alleles can also drive these patterns.

Background selection

Populations experience a constant influx of deleterious mutations at functional loci while selection acts to purge them from the population, thus preventing deleterious substitutions and maintaining function at these loci. As we discussed in Chapter \ref{Chapter:OneLocusSelection}, this balance between mutation and selection results in a constant level of deleterious variation in the population. The constant selection against this deleterious variation has effects on diversity at linked sites. Each deleterious mutation arises at random on a haplotype in the population, and as selection purges this mutation, it removes with it any neutral alleles that were also on this haplotype. This constant removal of linked alleles from the population acts to reduce diversity in regions surrounding functional loci , an effect known as background selection (BGS).

What proportion of our haplotypes are free of deleterious mutations in any given generation, and so free to contribute to future generations? Well, under mutation-selection balance, a constrained locus with a mutation rate \(\mu\) towards deleterious alleles that experience a selection coefficient \(sh\) against them in heterozygotes, will result in \(\frac{\mu}{sh}\) chromosomes carrying the deleterious allele. Some of these haplotypes may be passed on to the next generation, but if they are fully linked to the deleterious locus they will all eventually be lost because they carry a deleterious mutation at a site under constraint. Thus, for a neutral polymorphism completed linked to a constrained locus, only \(2N(1-\frac{\mu}{sh})\) alleles get to contribute to future generations. Therefore, the level of pairwise diversity in a constant population due to BGS at such a locus will be

\[\E[\pi] = 2 \mu \times 2N(1-\frac{\mu}{sh}) = \pi_0 (1-\frac{\mu}{sh})\]

where \(\pi_0= 4N\mu\), the level of neutral pairwise diversity in the absence of linked selection.

The effects of background selection are more pronounced in regions of low recombination, where neutral alleles are less able to recombine off the background of deleterious alleles. Thus, under background selection, we also expect to see reduced diversity in regions of lower recombination.

For a neutral locus that is a recombination fraction \(r\) away from a locus subject to constraint, the level of diversity is

\[\E[\pi] = \pi_0 \left(1-\frac{\mu sh}{2(c+sh)^2} \right) \label{eqn:pi_loc_BGS_1}\]

As we move away from a locus experiencing purifying selection, we increase \(c\), and diversity should recover. For example, moving away from genic regions in the maize genome we see the average level of diversity recover. This occurs in both maize and teosinte, the wild progenitor of maize. The dip in diversity around non-synonymous sites is stronger in teosinte, perhaps because the accelerated drift due to the bottleneck in maize may have somewhat released constraint on sites where very weakly deleterious alleles segregated previously at mutation-selection balance.

More generally, if a neutral locus is surrounded by \(L\) loci experiencing purifying selection at recombination distances \(c_1,\cdots,c_L\), then compounding Equation \ref{eqn:pi_loc_BGS_1} across these loci, the expected reduced diversity is approximately

\[\E[\pi] = \pi_0 \prod_{i=1}^L \left(1-\frac{\mu sh}{2(c_L+sh)^2} \right) \approx \exp \left( \sum_{i=1}^L \frac{\mu sh}{2(c_i+sh)^2} \right) \label{eqn:pi_loc_BGS}\]

To model an average neutral locus in a genomic region with a given recombination rate, we can imagine that our neutral locus is situated in the center of a large region with total recombination rate \(C\) and total deleterious mutation rate \(U\), where \(U = \mu L\). Then our expression for diversity, Equation \ref{eqn:pi_loc_BGS}, simplies to

\[\E[\pi] \approx \pi_0 \exp \left( \frac{-U}{(sh+C)} \right) \approx \pi_0 \exp \left( \frac{-U}{C} \right). \label{eqn:GW_BGS_1}\]

In this last approximation, we assume that we’re looking at a large region, with \(C \gg sh\) . Note that much like genetic load, Equation \ref{eqn:mut_load}, this expression depends only on the total deleterious mutation rate. Any dependence on the selection coefficient drops out, as weakly selected mutations segregate in the population at higher frequencies, but are also removed from the population more slowly, allowing more of the genome to recombine off the deleterious background.

For a first go at fitting this to genome-wide data, we could look at diversity in windows of length \(W\) bp (as in Figure \(\PageIndex{16}\)). If we assume that there is a constant rate of deleterious mutation per base pair, \(\mu_{BP}\), then \(U=\mu_{BP}W\). Furthermore, if our genomic window has a recombination rate \(c_{BP}\) per base-pair, our total genetic length is \(R=c_{BP}W\). Making these substitutions in Equation \ref{eqn:GW_BGS_1}, our window size cancels out to give

\[\E[\pi] \approx \pi_0 \exp \left(\frac{-\mu_{BP}}{c_{bp}} \right) \label{eqn:GW_BGS_2}\]

Looking across windows that vary in their recombination rate, i.e. \(c_{BP}\), we can fit Equation \ref{eqn:GW_BGS_2} to data to estimate \(\mu_{BP}\). An example of doing this to data from D. melanogaster is shown in Figure \ref{fig:GW_BGS_reduction}, yielding an estimate of the deleterious mutation rate of \(\mu_{BP}\approx 3.2 \times 10^{-9}\). This is roughly on the same order as the mutation rate per base pair in D. melanogaster, and so this deleterious mutation rate estimate is somewhat high as it would require most of the genome to be constrained, but as a first approximation it’s not terrible. Note how similar the fit is to a model of hitchhiking, suggesting that some combination of BGS and hitchhiking can explain the broad relationship between diversity and recombination seen in D. melanogaster and other species.

As our annotations of functional regions of the genome have improved, so have our methods to infer background selection. A more rigorous version of this analysis today would incorporate variation in coding density among windows into the parameter \(\mu_{BP}\). With detailed genomic annotations showing coding regions and constrained non-coding regions, we can also move beyond just analyzing broad-scale patterns. For example, fit a model of background selection to putatively neutral pairwise diversity along the human genome, using Equation \ref{eqn:pi_loc_BGS} to estimate the effect of BGS at each locus, weighing the genetic distance to all of the surrounding coding regions and constrained non-coding sites. This allowed to estimate mutation rates and average selection coefficients acting against deleterious alleles in these regions of the genome. This best fitting model also allowed them to predict diversity levels along the genome, a section of which is shown in Figure \(\PageIndex{17}\). Thus, broad-scale features of polymorphism along the genome are well described by background selection (or by linked selection more generally).

The deleterious mutation rates estimated by from fitting a model of BGS were again too high, as in the Drosphila example above, suggesting the BGS alone is not sufficient to explain all of the effect of linked selection. But how then do we go about distinguishing the impact of BGS from hitchhiking?

Distinguishing the impact of hitchiking from background selection in genome-wide data

A variety of approaches have been taken to start to separate the effects of hitchhiking from background selection. Much of the strongest evidence showing the effects of both comes from Drosophila melanogaster and we review some of that evidence here. Hitchhiking is expected to have systematic effects on the neutral site frequency spectrum, distorting it towards rare minor alleles, (reflecting the slow recovery of diversity following a sweep). Therefore, we should expect a distortion of summary statistics such as Tajima’s D in regions of low recombination if hitchhiking is contributing to the reduction in diversity in these regions . In D. melanogaster, there is a greater skew towards rare alleles at putatively neutral sites in regions of low recombination ; see left panel of Figure \(\PageIndex{18}\). However, while this skew isn’t expected under simple models of strong background selection.

Another prediction of the hitchhiking model, where an allele sweeps to fixation, is that there should be a functional substitution associated with each sweep. Or, to flip that around, we might expect to see a greater impact of hitchhiking where there are more functional substitutions. For example, regions surrounding non-synonymous substitutions should have lower levels of diversity, if a high fraction of non-synonymous substitutions are adaptive. Again, this pattern is seen in D. melanogaster .

Pushing this idea further, we can look at the dip in diversity surrounding a non-synonymous substitution averaged across all the substitutions in the genome. found a stronger dip in diversity around non-synonymous substitutions than synonymous substitutions . Extending the model of to fit a model of background selection and hitchhiking to putative neutral diversity along the genome, they found that the dip in diversity around synonymous substitutions comes mostly from BGS. But to fully explain the dip in diversity around non-synonymous substitutions, a reasonable proportion of these non-synonymous substitutions have to have been accompanied by a classic (hard) sweep. The majority of these sweeps are estimated to be due to very weak selection, with selection coefficients \(<10^{-4}\). Furthermore, estimated a \(77\) - \(89\%\) reduction in neutral diversity due to selection on linked sites, and concluded that no genomic window was entirely free of the effects of selection. Thus linked selection has a profound effect in some species such as Drosophila melanogaster.