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Genetics. Sep 2008; 180(1): 301–316.
PMCID: PMC2535683

Hitchhiking Both Ways: Effect of Two Interfering Selective Sweeps on Linked Neutral Variation


The neutral polymorphism pattern in the vicinity of a selective sweep can be altered by both stochastic and deterministic factors. Here, we focus on the impact of another selective sweep in the region of influence of a first one. We study the signature left on neutral polymorphism by positive selection at two closely linked loci, when both beneficial mutations reach fixation. We show that, depending on the timing of selective sweeps and on their selection coefficients, the two hitchhiking effects can interfere with each other, leading to less reduction in heterozygosity than a single selective sweep of the same magnitude and more importantly to an excess of intermediate-frequency variants relative to neutrality under some parameter values. This pattern can be sustained and potentially alter the detection of positive selection, including by provoking spurious detection of balancing selection. In situations where positive selection is suspected a priori at several closely linked loci, the polymorphism pattern in the region may also be informative about their selective histories.

THE search for molecular signatures of positive selection has been a matter of intense research and applications in the recent years, motivated by the hope to better understand the genetic bases of adaptation and the recent history of populations (Bamshad and Wooding 2003; Nielsen et al. 2007). The footprints of positive selection on neutral polymorphism are the consequence of the hitchhiking effect (Maynard Smith and Haigh 1974), and current methods to detect them encompass two main approaches. The first one is genome scans of neutral variation and is a top-down process. It consists of gathering polymorphism data widely distributed throughout the genome and summarizing them with a particular measure, be it the nucleotide diversity, the frequency spectrum of mutations (Nielsen et al. 2005), or the length and frequency of haplotypes [for ongoing selective sweeps (Sabeti et al. 2002; Voight et al. 2006)]. The loci exhibiting extreme values in the distribution of the measure are then considered as putative targets of positive selection (but see Teshima et al. 2006 for caveats of this method). The second approach, the candidate-gene approach, is a bottom-up process in which one wishes to test some evolutionary scenarios, for instance, for a gene (or QTL) of known function (see, for instance, Edelist et al. 2006). It consists of analyzing neutral polymorphism at a finer scale (of the order of the megabase or lower), to test if positive selection occurred, and to infer some parameters of the selective sweep such as the target and strength of selection. This fine-scale analysis can also be carried out in regions identified after a genome scan (a good example of this kind is Pool et al. 2006; for a more comprehensive review see Thornton et al. 2007). Here, we focus on this finer-scale analysis of polymorphism.

The most popular method for the fine-scale analysis of selective sweeps uses the information at several markers distributed in the small region of interest, to perform a composite likelihood-ratio test on the frequency spectrum (Kim and Stephan 2002), to jointly estimate the parameters of the selective sweep and the relative likelihood of selection vs. neutrality. This can be followed by a goodness-of-fit test to confirm the robustness of the estimated parameters against several demographic scenarios (Jensen et al. 2005). Though efficient, this method can be affected by ascertainment biases (Thornton and Jensen 2007). Moreover, some factors—e.g., differences in recombination or mutation rates between the two sides of a selective sweep—can modify the fine-scale polymorphism pattern around the selective sweep in a systematic way (i.e., nonstochastically). Here, we focus on one particular modifying factor, namely the presence of another locus under positive selection in the region of influence of a selective sweep. We wish to understand how the effect of a focal selective sweep is modified by the presence of another selective sweep in its vicinity.

Simultaneous positive selection at several linked loci was repeatedly reported for asexuals (Notley-McRobb and Ferenci 2000; Perfeito et al. 2007), where it was termed “clonal interference.” The effect of such interference on probabilities of fixation in asexuals was described theoretically by Gerrish and Lenski (1998). In sexuals, positive selection at two closely linked loci not only decreases their probabilities of fixation (Barton 1995), but also builds up negative linkage disequilibrium between them (Hill and Robertson 1966; Felsenstein 1974) and slows down their dynamics, which altogether is called the Hill–Robertson effect. Notably, the overlap in time of positive selection at partially linked loci, with (Barton 1995) or without (Roze and Barton 2006) epistasis, is invoked in all population genetic models of the evolution of sex. This phenomenon is difficult to characterize empirically in natural populations. One of the reasons is that selection at two closely linked loci may be difficult to detect through its signature on neutral polymorphism without a priori information, since the signatures of both loci may be confounded. Moreover, the lack of knowledge about the effect of two interfering selective sweeps on neutral polymorphism makes it difficult to look for such signatures. Yet, in cases where one a priori suspects recent selection at two closely linked loci, signatures of selection can be found. This was done in two recent studies. The first one concerns two genes involved in sex-ratio distortion in Drosophila simulans (Derome et al. 2008). The second one deals with the domestication gene Tb1 and the early-flowering gene dwarf8 in maize (Camus-Kulandaivelu et al. 2008). These two studies at least suggest that successful selective sweeps at two tightly linked loci can occur in natural populations.

Some models where several selective forces interact on neutral polymorphism were published in the last decade. Kim and Stephan (2000) investigated the joint effects of positive and negative selection on neutral polymorphism and showed that the hitchhiking effect dominates in regions of low recombination, whereas background selection primarily explains the levels of neutral heterozygosity in regions with higher recombination. Kim and Stephan (2003) studied the hitchhiking effects of two selective sweeps that overlap in time, that is, the interplay of positive selection at two loci. Their aim was mainly to assess whether predictions made under the assumption that selective sweeps do not overlap still hold when there is at least partial overlap. They showed that because of the selective interference between the loci under selection, (i) their time to fixation increases, which leaves more time for recombination with the neutral locus to occur, and (ii) the probability of fixation is decreased for each beneficial mutation. The net effect is an overall decrease of the effect of selective sweeps relative to the case without interference.

Here, we further study the interplay of positive selection at two closely linked loci, with a different perspective. We focus on cases in which both beneficial mutations escape stochastic loss and get fixed and ask what the resulting pattern of neutral polymorphism is in the region. We want to know how a successful selective sweep at a linked locus alters the signature of a focal selective sweep on heterozygosity and on the site-frequency spectrum. We also investigate whether or not there is a particular signature of the action of two close selective sweeps and whether neutral polymorphism can carry information about the history of adaptive selection at two loci. We show that the interference of two selective sweeps can dramatically affect the signatures of positive selection, in particular by inducing an excess of intermediate-frequency variants in the frequency spectrum. This may paradoxically hinder our ability to detect adaptive selection in regions of the genome where it was most experienced.


Let us first use a deterministic argument to introduce the problem. We want to calculate the change in allelic frequencies at a neutral locus neu under the influence of hitchhiking effects from two loci under positive selection, sel1 and sel2, with selection coefficients s1 and s2, respectively. We assume that all loci are biallelic. The frequencies of the beneficial alleles at sel1 and sel2 are denoted equation M1and equation M2, respectively, and we denote pneu the frequency of an arbitrarily chosen neutral allele at neu. The recombination rate between any pair of loci {l, m} is denoted rl,m. The fitness of an individual carrying equation M3 copies of the beneficial allele at sel1 (equation M4) and equation M5 (equation M6) copies of the beneficial allele at sel2 is

equation M7

that is, we assume that fitness is additive within each locus and multiplicative between loci. The changes in frequencies at all loci due to selection at both sel1 and sel2, as well as other relevant parameters, were derived by exact recursions (see appendix a).

We denote CU the linkage disequilibrium between a locus set U, defined as the covariance of their allelic states as in Barton and Turelli (1991) (see appendix a). The changes in frequencies at the selected loci can be written in a general form as

equation M8

where seli stands for the focal selected locus (sel1 or sel2) and selj stands for the other selected locus (sel2 or sel1, respectively), and equation M9. The change in frequency at the neutral locus is

equation M10


equation M11

Assuming equation M12and equation M13, such that the terms involving products of the selection coefficients can be neglected, Equation 2 can be rewritten:

equation M14
equation M15

The approximate Equation 3a shows that when selection is weak, the frequency of the beneficial allele at a selected locus seli changes not only because of its own effect on fitness (equation M16), as if it was alone, but also because of the hitchhiking with the other selected locus selj (equation M17), which depends on the other selection coefficient and the linkage disequilibrium, as if seli was neutral. This latter term captures the interference between the selected loci. At the neutral locus, when selection is weak the change in frequency is simply the sum of the effects of hitchhiking with both selected loci, each of which depends on the respective selection coefficient and on the linkage disequilibrium between the neutral locus and the locus under selection (3b).

At this stage, we can get a first feeling of the dynamics of the neutral allele frequency based on Equation 3b. If equation M18 and equation M19 have the same sign, then the two selective sweeps have cumulative effects on the change in frequency at the neutral locus: the hitchhiking effects are synergistic, and the neutral allele frequency will evolve faster than if it was hitchhiking with only one selected allele (“single selective sweep”). In contrast, if equation M20 and equation M21 have opposite signs, the hitchhiking effects are antagonistic and will tend to compensate each other. In such cases, the frequency of a neutral allele exposed to a “double” selective sweep may evolve slower than that under a “single” selective sweep.

The intensity of the interaction (synergy or antagonism) of hitchhiking effects depends on the selection coefficients as well as on the magnitudes of the linkage disequilibria equation M22 and equation M23, so in the long run, the change in frequency at the neutral locus also depends on the dynamics of equation M24 and equation M25. The initial values of those linkage disequilibria vary with the starting conditions. Their exact dynamics during the sweeps are too complicated to give here (see appendix a). However, to a first-order approximation with small s1 and s2, the recursions for two- and three-locus linkage disequilibria read

equation M26
equation M27
equation M28

with equation M29, where loci A, B, and C represent the previously defined loci (neu, sel1, sel2) ordered along the chromosome (see appendix a). In other words, equation M30 is the probability that there is no recombination between the “first” and the “second” locus, and no recombination between the second and the “third” locus, where first, second, and third refer to the position of the loci on the chromosome, regardless of their status (neutral or selected). The linkage disequilibrium between the selected loci, equation M31, either increases or decreases as a result of selection, depending on the sign of equation M32, and systematically decreases (in absolute value) with increasing recombination. Our main focus in our examination of the hitchhiking effect is the linkage disequilibria between the neutral locus and each of the selected loci, equation M33 and equation M34, as shown in Equation 3b. For each selected locus seli, equation M35 is modified because of selection at seli and also because of selection at the other selected locus selj, by the means of the three-way linkage disequilibrium equation M36 (Equation 4b).

Hence, the selective interference between sel1 and sel2 affects the variation of the neutral allele frequency through the higher-order interaction, i.e., the three-locus linkage disequilibrium (Equation 4b). This makes sense and gives an illustration of the often difficult to understand meaning of three-locus linkage disequilibrium. When there is selective interference between sel1 and sel2, the linkage disequilibrium between these two selected loci does not directly affect the neutral allele frequency. However, regardless of the association between sel1 and sel2, a nonequilibrium repartition of neutral alleles among two-locus haplotypes at sel1–sel2 does affect the dynamics of linkage disequilibria equation M37 and equation M38 (Equation 4b), which in turn influences the hitchhiking effects; three-locus linkage disequilibrium is indeed a measure of this repartition. Note that the influence of selective interference through the three-locus linkage disequilibrium is directly apparent from the change in frequency at the neutral locus (Equation 2b) but should have an effect only when selection is strong. Again, recombination always decreases (in absolute value) the association between neu and seli; therefore the location of neu is critical to the outcome of the double selective sweeps. If neu is inside the interval delimited by sel1 and sel2, the difference between equation M39 and equation M40 is always smaller than if neu were outside of the interval. The difference between the rates of variation of equation M41 and equation M42 is then larger when the neutral locus is outside the interval. Moreover, for a given distance between sel1 and sel2, equation M43 is always smaller when the neutral locus is outside of the interval than when it is inside. Hence, when neu is inside the interval, the interplay of hitchhikings from both selected loci (whether synergy or antagonism, depending on the starting conditions) is more likely to be sustained. In contrast, if neu is outside this interval, in the long run the hitchhiking effect by the closest selected locus dominates, and there is less opportunity for either antagonism or synergy of hitchhikings effects.

For the rest of this article, we define neum as the neutral locus located exactly in the middle of the interval delimited by sel1 and sel2 and neui as the neutral locus located at the same distance from seli as neum, but on the other side of seli, outside of the interval. The recombination rates with the loci under selection are thus, for neum, equation M44 (by definition), and for neui (i = 1 or 2), equation M45 (by definition) and equation M46. Hence, for neum, the linkage disequilibrium with the two loci under selection is similarly affected by recombination, whereas, for neu1 and neu2, the linkage disequilibrium with the farthest selected locus is three times more affected by recombination as the one with the closest selected locus. As a consequence, under similar strength of selection at both loci, we expect the neutral polymorphism at neum to reflect the interplay of selection at the two loci, whereas the polymorphism at neu1 and neu2 will carry mainly the signature of selection at the closest selected locus.

Apart from the dynamics of the selected loci and the changes in linkage disequilibria, the type of interaction between the selective sweeps (antagonism or synergy) strongly depends on the initial conditions. In the simplest case where both beneficial mutations enter the population at the same generation, they are most likely in negative linkage disequilibrium, and the probability that they are associated with different alleles at a neutral locus equals the heterozygosity at that generation (i.e., the probability of drawing two different alleles at a locus). In a given fragment of sequence, there are several polymorphic sites, for which the frequency distribution of mutant alleles is well known (Ewens 2004). Thus, during one occurrence of a double selective sweep, several initial conditions regarding the initial linkage disequilibria equation M47 are encountered among the various polymorphic sites. As a consequence, the double selective sweep affects not only the global neutral diversity (as measured by the heterozygosity or nucleotide diversity), but also the repartition of this diversity among sites (as measured by the frequency spectrum of mutations). Moreover, the initial frequencies of the neutral mutations dramatically influence their evolutionary dynamics and their final frequency at the end of the sweep. The frequency spectrum is then a key indicator here, and we wish to describe its evolution under the influence of two interfering selective sweeps, by following small sequence fragments located in the vicinity of the selected loci. Moreover, we wish to know if the general processes that we described using the deterministic model still matter when starting from a realistic initial distribution of neutral allelic frequencies.

Since the changes in frequencies at many tightly linked polymorphic sites are not analytically tractable, we used Monte Carlo simulations to address this question. This also allowed us to take into account the stochasticity inherent to every actual population, which may have important consequences in several aspects of the process. For instance, in a finite population, the two selected mutations need to end up on the same haplotype (by recombination) to both get fixed (Hill and Robertson 1966). This represents a qualitative shift that cannot appear in an analytical treatment and yet strongly influences the outcome of the selective sweeps. Moreover, forward Monte Carlo simulations allowed us to stochastically introduce new polymorphic sites through mutation during the selective sweep [infinite-site model (Ewens 2004)]. Finally, by simulating the sampling of gametes by the experimenter, we could include the sampling variance in our analysis. In the following, we present the approach and results of our forward simulations of interfering selective sweeps.


Forward simulations:

We used forward individual-based Monte Carlo simulations to investigate the effects of selection at two closely linked loci on neighbor neutral polymorphism. We simulated polymorphism at several sequence fragments along a chromosome region encompassing two sites under positive selection. Each fragment evolved under the infinite-site model of mutation. Recombination was allowed within and between fragments. The actual number of sites in each fragment was not explicitly defined; instead, a continuous model was used, in which the mutation parameter θ = 4Neμ and the recombination parameter ρ = 4Ner (where Ne is the effective population size) were defined at the level of the entire fragment.

At the beginning of each simulation, the initial conditions were settled for each fragment by generating the whole population by coalescence using the program “ms” (Hudson 2002). This provided realistic initial conditions regarding the distribution of polymorphism in each fragment, without having to simulate the complex genealogical relationships between the fragments, since we did not wish to measure the linkage disequilibrium between fragments. Although coalescence theory is generally used for samples that are small relative to the population size, Wakeley and Takahashi (2003) showed that when the sample size equals the effective population size, the error induced by using the coalescent is minute. Indeed, by neglecting multiple coalescence events, the standard coalescent expectation underestimates by ~12% the expected number of alleles present in a single copy in the entire population, all other frequency classes remaining essentially unchanged. We checked that this approximation did not affect our results by artificially increasing the proportion of singletons by 12% in the initial population generated with ms and then running the forward simulations. The outcome was equivalent to that without increasing the number of singletons, thus validating the accuracy of our method (results not shown). We used θ = 5 and ρ = 10 as parameters, such that the ratio ρ/θ was similar to that documented for Drosophila (Kliman et al. 2000; Przeworski et al. 2001).

We considered two selected loci: sel1 and sel2. The selective phase was simulated forward in time. It started with the introduction of the beneficial allele at sel1 and ended when the beneficial alleles at sel1 and sel2 were both fixed. If any of the beneficial alleles was lost before fixation, the run was discarded and a new simulation was started again with the same initial conditions. For each locus under selection, the haplotype carrying the beneficial allele was introduced in five copies. This reduced computing time by lowering the risk that a beneficial allele was lost by drift in the early generations. This procedure is justified since our observations are conditioned on the final fixation of both mutations. Indeed, according to Barton (1998), conditional on its final fixation, a beneficial mutation rises quickly in frequency in early generations, and thus there is negligible opportunity for mutation or recombination to occur on the haplotype that carries it. In practice, for each selected locus, a single haplotype from ms was copied five times and the beneficial mutation was placed on it. Hence this approach was meant to model the rapid increase in frequency of the beneficial mutation in the early generations (conditional on fixation). It should not be confused with a selective sweep from the standing variation, where a mutation first drifts neutrally for several generations and then becomes selected when it is at a frequency >1/(2N). In such a “soft” selective sweep, the beneficial mutation may initially be present on several distinct haplotypes. The neutral signature of such a soft sweep may be very different from that of a hard selective sweep, as Przeworski et al. (2005) showed, but this is not the topic of this article.

During the selective phase, mutation and recombination rates were defined at the individual rather than the population level, using the same μ and r as in the neutral phase. For each fragment, when mutation occurred in a gamete, it was simulated by randomly drawing a position inside the fragment out of a continuous uniform distribution and introducing a derived allele at this position. Recombination was simulated in the same manner between the neutral fragments and the sites under selection, as well as inside the fragments, using Haldane's mapping function assuming no interference.

Signatures of selection:

At the end of the simulation, several measures were made. First, we computed the reduction of heterozygosity in the entire population. This was expressed as the ratio π/πo of the observed nucleotide diversity per fragment over its value at the beginning of the selective phase. We also simulated the sampling of a small number of individuals (2n = 50 gametes) to assess the properties of the frequency spectrum and to perform some tests of selection. The samples were drawn conditionally on the presence of polymorphism in at least one fragment. The frequency spectrum was calculated as in Kim (2006). We computed the proportions of sites belonging to each frequency class (i.e., from 1 to 2n − 1) for each repeat and then averaged these proportions over all the repeats. For each simulation run, we also calculated several summary statistics for the frequency spectrum of mutations. The first one, Tajima's D (Tajima 1989), is the normalized difference between Watterson's (1975) estimator of θ based on the number of polymorphic sites and Tajima's (1983) estimator π based on the heterozygosity of sites. A negative value denotes an excess of low-frequency variants, indicative of positive selection or of population expansion, whereas a positive value denotes an excess of intermediate frequencies as can be caused by balancing selection. When necessary, we also calculated the H statistics defined by Fay and Wu (2000), in the standardized version proposed by Zeng et al. (2006). A negative value is indicative of an excess of very-high-frequency variants, which is a signature of positive selection. Finally, we used Zeng et al.'s (2006) E statistics, which contrast the abundance of low- vs. high-frequency variants. We report the mean values of these statistics over 500–1000 runs of simulations. We also assessed their respective powers to reject neutrality. This was calculated as the proportion of simulations for which the value of the statistics led to rejecting neutrality at the 5% significance level. Significance was assessed by running 10,000 coalescence simulations with ms (Hudson 2002) with the same sample size and the same number of polymorphic sites as the mean of selection simulations. We used the subset of ms samples in which at least one polymorphic site was present, as in Przeworski (2002). All statistics were used in one-sided tests, including Tajima's D, again as in Przeworski (2002). This means that we considered that positive and negative values of Tajima's D bear distinct information and lead to different evolutionary interpretations. As such they can be treated as different statistical tests instead of just being pooled as a global rejection of neutrality.


The symmetric case:

We first consider the case in which mutations at sel1 and sel2 appear simultaneously in the population and have the same selection coefficients (s1 = s2 = s = 0.1). Though this is likely not the most realistic situation, we use it as a case study to better understand and illustrate the various forces in action. As this situation is completely symmetric, we can consider only one-half of the chromosome segment, namely the sel1 side from neu1 to neum. In the following, neu1 and neum do not refer to specific polymorphic sites, but rather to small chromosomal regions centered on the position described earlier for these loci.

Reduction of heterozygosity:

Figure 1A shows the reduction of heterozygosity (quantified by π/πo) at neutral markers along the chromosome. One selected locus (sel1) is at position 0, and the other selected locus (sel2) is on the right-hand side at +0.5 cM from sel1 (not shown). The neutral locus at the right end of the graph is neum, located in the very middle of the interval [sel1–sel2]. Hence, the right side of the graph (positive abscissa values) gives the pattern for “inside” neutral markers located in the interval [sel1–neum], and the left side of the graph (negative abscissa values) gives the pattern for “outside” neutral markers located in the interval [telomere–sel1]. In the symmetrical case, the pattern on the right of neum is symmetrical (not shown). The solid line in Figure 1A gives the pattern of heterozygosity when selection acts at both loci sel1 and sel2 (hereafter we refer to this case as “double selective sweep”). As a comparison, the dashed line gives the pattern when there is selection at only one locus, here at sel1 (“single selective sweep”). On the graph, the neutral loci on the left side (neu1) and the right side (neum) are at the same distance from sel1. This is used to compare inside and outside loci for the same recombination rate with the selected locus and also to compare single and double selective sweeps (see below).

Figure 1.
Polymorphism patterns along the chromosome. (A) Reduction of heterozygosity (π/πo) in the whole population; (B) Tajima's D in a sample of size 2n = 50 chromosomes, as a function of the distance to sel1. Solid line: selective sweeps ...

The pattern observed in Figure 1A for a single sweep (dashed line) is the well-known classical picture (Maynard Smith and Haigh 1974; Stephan et al. 1992; Kim and Stephan 2002) except that here the diversity is not zero for a marker located at position zero (i.e., on the selected locus sel1), because mutation takes place during the course of the selective sweep in our forward simulation model. For a double selective sweep (solid line), the pattern on the left of sel1, i.e., outside the selected bracket, is very similar to that obtained for a single selective sweep at sel1 of the same intensity. In contrast, between sel1 and sel2, i.e., inside the selected bracket, the neutral polymorphism is substantially higher than outside the bracket or than the case of a single selective sweep.

This is the first main result of the simulations, consistent with the deterministic explanation above. In the case where there is interference between selective sweeps of similar intensities at two linked mutations, the pattern of polymorphism outside the selected bracket resembles that of a single selective sweep; i.e., even when there is selection at both sel1 and sel2, neutral loci on the left of sel1 are mostly affected by selection at sel1, not at sel2. In contrast, for neutral markers lying between the selected loci, the diversity at the end of the selective sweeps is the result of the combined effects of both hitchhikings. In particular, in the case of antagonistic selective sweeps that start at the same generation at sel1 and sel2, more polymorphism is maintained than in the case of a single selective sweep of the same intensity.

Frequency spectrum and Tajima's D:

The frequency spectra at neu1 and neum in a sample of size 2n = 50 are shown in Figure 2. At neu1, the spectrum is characterized by an excess of high-frequency derived variants, a lack of intermediate-frequency variants, and an excess of low-frequency variants relative to the neutral expectation. Taken together, these features are typical of a neutral locus partially linked to a locus under positive selection (Tajima 1989; Fay and Wu 2000; Przeworski 2002). At neum, there is an excess of high-frequency variants, consistent with positive selection, but a lack of low-frequency variants. More importantly, there is an excess of variants at intermediate frequencies (from 15 to 35) relative to the standard neutral case. Taken alone, this latter feature is commonly interpreted as the outcome of selective forces maintaining diversity, i.e., balancing selection (Charlesworth 2006). However, taken together the features of the frequency spectrum at neum—excess of high-frequency derived variants, excess of intermediate-frequency variants, and lack of low-frequency variants relative to the neutral expectation—appear as a distinctive pattern of a double selective sweep.

Figure 2.
Frequency spectrum of mutations in a sample of size n = 25 diploid individuals (2n = 50 chromosomes). Solid line, expectation under standard neutrality; open bars, neu1 (outer locus); shaded bars, neum (inner locus). Results are averaged ...

To understand how the neutral-frequency spectrum changes along the chromosome in the case of a double selective sweep, we used Tajima's D as a summary statistic because it is the most sensitive to perturbations of intermediate-frequency classes (Tajima 1989; Przeworski 2002). Figure 1B shows the pattern of Tajima's D along the chromosome, with the same formalism as in Figure 1A. For the single selective sweep (dashed line), Tajima's D is negative in the entire region considered, as a consequence of the lack of intermediate-frequency variants generated by the strong positive selection. Note that the values are higher for the marker that includes the target of selection than for markers in the close flanking regions, as a consequence of the smaller number of polymorphic sites in the former than in the latter. In the case of a double sweep (solid line), Tajima's D is globally higher than for a single selective sweep, even though the selection coefficient at sel1 is the same. Moreover, the pattern is strongly asymmetric: Tajima's D inside the selected interval is higher than outside the interval. In the case of Figure 1B, Tajima's D is even positive in the middle of the selected interval. This is the second important result of this article, which was not directly predictable from the deterministic model above: the interference of two selective sweeps has more impact on the frequency spectrum than on the reduction of heterozygosity. In our illustrative example, neum exhibits a reduced heterozygosity, which is consistent with positive selection in the region (Figure 1A, right edge of the solid line), whereas Tajima's D, which summarizes the frequency spectrum, does not carry any signature of positive selection and is even positive (Figure 1B, right edge of the solid line).

The frequency spectrum is obviously affected by the stochasticity inherent to the finite population size and to the sampling, so there is variation in Tajima's D between repeats, which we report in Table 1. For the sake of simplicity we show only results for neum and neu1, as well as for a neutral locus equivalent to neu1 in the case of a single selective sweep. The standard deviation of Tajima's D in case of a double selective sweep is larger than that for a single sweep of the same intensity. Moreover, the standard deviation at neum is much larger than that at neu1. Indeed, neum is more directly under the combined effects of two hitchhikings than neu1, so slight changes in the starting conditions can lead to more variation in the final state at neum than at neu1.

Variation pattern of Tajima's D

We also report in Table 1 (columns 3 and 4) the power to reject neutrality in our simulations, using Tajima's D in one-sided tests (see methods). Interestingly, at neum it is substantially more probable to reject neutrality through a significantly positive value (>27% of the simulations) than through a significantly negative value. This is not true for neu1, for which the powers of the tests on the right side and on the left side are comparable to those for a single selective sweep of the same intensity.

Area of influence:

We explored the range of recombination values between the selected loci for which the selective sweep at sel2 had an influence on the polymorphism pattern generated by the selective sweep at sel1. This was done by increasing the distance between sel1 and sel2, while keeping the selection coefficients constant, and relocating the neutral markers such that neum remained in the middle of the interval and neu1 lay outside the interval at the same distance from sel1. The results are shown in Figure 3, where the reduction in heterozygosity (π/πo) at neum and neu1 is plotted as a function of the ratio equation M50 between recombination and the selection coefficient. Interestingly, neum remains more polymorphic than neu1 for values of equation M51 spanning more than two orders of magnitude (Figure 3A). This range corresponds to that usually documented for the region of influence of a selective sweep (Fay and Wu 2005). Again, the frequency spectrum is more sensitive than the polymorphism level to the interference between selective sweeps (Figure 3B). For instance, at equation M52, neum and neu1 have similar reduction in heterozygosity but different Tajima's D values, with a slightly positive value at neum. The strongest asymmetry in Tajima's D between neu1 and neum occurs at equation M53 (i.e., when the recombination rate between the neutral locus and the locus under selection is such that r/s = 0.01). At this point, the difference between Tajima's D for neum and neu1 is 1.83. Note that as equation M54 increases, Tajima's D at the middle of the interval has reduced positive values, but those extend over a larger chromosomal region.

Figure 3.
Influence of the distance between selected loci. (A) Reduction of heterozygosity in the whole population; (B) Tajima's D at neum (solid line) or neu1 (dashed line), in a sample of size 2n = 50, plotted against the distance between the selected ...

Duration of the signature:

The footprints left by selective sweeps on neutral variation are obviously transient, since mutation and drift eventually restore the heterozygosity and the frequency spectrum to their neutral equilibria. The duration of such a signature is a key issue in the detection of selection in natural populations and has been recently a subject of much interest (Przeworski 2002, 2003; Jensen et al. 2005). In our case, we wished to know how the particular pattern of polymorphism induced by a double selective sweep evolved after the end of the sweeps. Figure 4 shows the power to reject neutrality after the fixation of both beneficial mutations, using three summary statistics for the frequency spectrum: Tajima's D, Fay and Wu's H, and Zeng's E (see methods). After the end of the selective sweeps, the power of Fay and Wu's H decreases much faster than that of Tajima's D (Figure 4, A and B), as was already discussed in Przeworski (2002). For all three summary statistics, the power to reject neutrality outside the sel1–sel2 interval remains higher than inside the interval, over the entire period considered. Zeng et al. (2006) emphasized that the power of their new statistic E increased after the end of the selective sweep (or bottleneck), coinciding with the decrease of H, so that these two statistics had somehow antagonistic behaviors because they both depended on high-frequency variants. Our simulation results show that after two simultaneous selective sweeps, the power of E increases more slowly for neum than for neu1. Thus E relays the information contained in H, including the differences between the powers to reject neutrality at loci located inside or outside the selected interval. Altogether, our results indicate that the signature left by antagonistic hitchhiking effects may persist for a long time after fixation of the beneficial mutations.

Figure 4. Figure 4. Figure 4.
Power to reject neutrality after the sweeps: proportion of simulations (of 500–1000 repeats) that reject neutrality at the 5%-significance level using (A) D, (B) H, or (C) E as summary statistics, plotted against the time T (in generations) since ...

Relaxing the symmetry:

Until now, we have focused on an illustrative, completely symmetric case, in which selective sweeps at sel1 and sel2 occurred simultaneously and where both mutations had the same selection coefficient. In practice, it is unlikely that two beneficial mutations arise at the same generation at two closely linked loci. Also, selection coefficients may vary importantly between beneficial mutations. The interaction between selective sweeps is expected to depend on the synchronicity of the beneficial mutation events at sel1 and sel2, as well as on their relative selection coefficients, which determine how long in time the sweeps will overlap. To assess the influence of these parameters on the final pattern of polymorphism, we ran simulations where the beneficial allele at sel1 appeared first and then was allowed to reach a threshold frequency pt before the beneficial allele at sel2 was introduced. The threshold frequency pt was transformed into a scaled time τ to account for the fact that the trajectory of a beneficial allele is not linear in time (see appendix b). The selection coefficient s1 was kept constant, while s2 was varied such that equation M57.

Figure 5 shows the resulting Tajima's D at neum, neu1, and neu2. At outside loci, neu1 and neu2 (Figure 5, A and C), the delay between selective sweeps has little influence on Tajima's D at fixation. At neu1, the impact of the hitchhiking by sel2 was substantial only when s2 > s1 and τ was close to 0, i.e., when the selective sweeps had little delay (Figure 5A, dashed line). At neu2 (Figure 5C), the final Tajima's D obviously depends on s2 as it is the selection coefficient of the closest selected locus, but the influence of selection at sel1 is weak except when s2 < s1. In contrast, at neum (Figure 5B), the frequency spectrum is under strong influence of both hitchhiking effects, and the outcome is highly dependent on the timing of the sweeps and on the ratio of the selection coefficients. When s2s1 (Figure 5B, thin and thick lines), Tajima's D is maximal at τ = 0, i.e., when the sweeps tend to be simultaneous, and decreases rapidly with increasing τ. When s2 > s1, there is a nonzero value of τ that maximizes Tajima's D (Figure 5B, dashed line). This is because the dynamics of the selected allele frequencies are different from each other; hence a delay can enhance the antagonism of hitchhiking effects by allowing the beneficial alleles at sel1 and sel2 (i) to enter synchronously in the critical phase of a hitchhiking effect, in which the dynamics of the beneficial allele is quasi-deterministic, while this allele remains at a low frequency (Barton 1998), and (ii) to reach nonnegligible frequencies at similar times, so that several recombination events can produce haplotypes hosting the two favorable alleles in coupling. Note that for this to happen, the weaker mutation must start increasing in frequency earlier, so it has more chances to escape loss by drift due to the interference with the stronger mutation (Barton 1995). Therefore, this scenario is also the most likely to be encountered in real data exhibiting fixation at both selected loci. As expected, in all three situations there is a value of τ for which Tajima's D becomes lower at neum than at outer loci, indicating that hitchhiking effects in the middle of the interval switch from antagonism to synergy. This occurs all the earlier (in time) when the second selective sweep has a lower selection coefficient, because selective interference between the beneficial mutations is then reduced.

Figure 5. Figure 5. Figure 5.
Delayed selective sweeps with varying selection coefficients: final Tajima's D at (A) neu1, (B) neum, and (C) neu2 as a function of the scaled delay τ between the introduction of the beneficial mutations at sel1 and at sel2 (see appendix b). Thin ...

Note that in nonsymmetric cases, we always kept neum at the middle of the selected interval, although antagonism between hitchhiking effects is not necessarily maximal at this point in the case of selective sweeps of different intensities. Hence, our simulation results at neum are conservative regarding the antagonism of hitchhiking effects.


Two selective sweeps of comparable intensities that arise close to each other tend to interfere in their effects on neutral variation. They can maintain globally higher levels of polymorphism than a single selective sweep of the same intensity, because selective interferences can slow down the dynamics of each mutation, allowing for more recombination (Kim and Stephan 2003). Here, we further show that, regardless of the trajectories of beneficial mutations in time, two interfering selective sweeps can compete in their hitchhiking effects simply by dragging along different neutral alleles. The resulting polymorphism pattern (as quantified by the nucleotide diversity π) is highly asymmetric, the region between the selected loci being the most subject to the interplay of the two hitchhiking effects. More importantly, we show that the interference of selective sweeps can distort the frequency spectrum in the direction of an excess of intermediate frequencies at neutral sites between the two selected loci, which is often interpreted as a signature of balancing selection (Charlesworth 2006).

Here, we conditioned the simulations on the fixation of both beneficial mutations. The actual fixation probability of beneficial mutations cannot be calculated directly from our simulations, since these mutations were introduced in several copies to decrease simulation time. The decrease in the probability of fixation as a consequence of selective interference was studied in detail in Barton (1995) and can be substantial. The conclusions of this study are thus more accurately applicable to cases where selection coefficients are large and of the same order of magnitude, for which the probability of joint fixation of both mutations is not negligible. Note that generally theoretical studies of selective sweeps based on coalescent simulations assume fixation of the beneficial mutation. Most of these studies also rely on the assumption that the product Ns is of the order of 500–1000 while the population size is very large (of order 106), such that the selection coefficient and the fixation probability are of order 10−3 (see, e.g., Fay and Wu 2000, 2005; Przeworski 2002). Also, interference of selective sweeps could well occur between beneficial mutations already present in the population and initially neutral, for instance, following a rapid environmental shift, which greatly decreases the risk of stochastic loss (Hermisson and Pennings 2005). Such selective sweeps from the standing variation are expected to leave a footprint different from that of a hard sweep (Innan and Kim 2004; Przeworski et al. 2005). Yet, since most neutral mutations are expected to be in low frequency in a natural population (Ewens 2004), it is quite possible that very few copies (if not a single one) of the beneficial mutation actually sweep through the population, hence turning the soft sweep into a quasi-hard sweep.

It may be argued that asymmetry in the polymorphism pattern may well arise by chance in a single selective sweep. Indeed, Figure 1 shows the mean of several simulations corresponding to an expected pattern, while obviously there is variation between repeats. Hence, some single-sweep simulations could exhibit a pattern similar to the one expected under interfering selective sweeps. Nevertheless, in the context of a candidate region where selection is searched for, the current practice is to use several markers distributed throughout the region. As the number of markers increases, it is less and less likely that an asymmetric pattern will be observed by chance for all the markers. For instance, in Figure 1, there are five markers on each side of sel1. Using |log(π(left side)/π(right side))| > 0.5 (where | | denotes absolute value) as a criterion for asymmetry for each couple of markers equally distant from sel1, the probability that asymmetry is in the same direction for all markers is 2.5 times higher under interfering selective sweeps than in the case of a single selective sweep. Palaisa et al. (2004) observed marked asymmetry at multiple markers in a genetic region. In such cases, a deterministic explanation might be involved rather than just chance variation, and the occurrence of a second interfering selective sweep should be considered together with other possible causes of asymmetry, such as differences in recombination or mutation rates between both sides of the selective sweep.

Santiago and Caballero (2005) showed that in a highly subdivided population, a selective sweep can induce an increase of heterozygosity and an excess of intermediate-frequency variants in demes other than that where the beneficial mutation originated. This is because the selective sweep can force the introduction of neutral alleles that were previously absent or in negligible frequency in those demes because of low migration. Our study of interfering selective sweeps can be understood in the light of their results by using an analogy in which the alleles at one selected locus define demes for the other selected locus, and recombination is viewed as “migration” from one genetic background to the other. This analogy is the rationale for the so-called “structured coalescent” approach of selective sweeps (Kaplan et al. 1989). In our context, the case where both beneficial mutations are initially in strong negative linkage disequilibrium and carry different neutral alleles is similar to that in Santiago and Caballero (2005), where a selective sweep starting in one deme hitchhikes an allele absent in another deme. Indeed, the focal selective sweep introduces neutral polymorphism in the other selected background, thus reducing the effect of the other selective sweep, and reciprocally. This illustrative analogy is not a mere equivalence, though, since in the case of interfering selective sweeps, the sizes of the “demes” change with selection. There is also selective interference between the selected loci, which alters the process by slowing down the dynamics at each locus, so our results are not redundant with those of Santiago and Caballero (2005).

We focused on interference between sweeps at reasonably distant selected loci, which results in the asymmetric pattern described in Figure 1, when the initial linkage disequilibrium is negative. In contrast, in cases where the beneficial mutations are too closely linked to recombine in a reasonable time (for instance, when they are inside the same gene), and yet have similar enough selection coefficients to be maintained at high frequencies for a long time, positive selection can contribute to maintaining high levels of nucleotide diversity very close to the target of selection. This situation is what was termed trafficking by Kirby and Stephan (1996). At the most extreme, several beneficial mutations could arise at the same site and several copies of the same allele, identical in state but not by descent, could provoke interfering selective sweeps. This was studied as a particular case of “soft sweeps” by Pennings and Hermisson (2006), who focused on the signatures left by selection at a site where a beneficial mutation was introduced recurrently by mutation during the course of the sweep. We believe that this study could contribute to generalizing the somehow extreme (though very enlightening) cases of soft sweeps with recurrent mutation and of “trafficking” to arbitrarily distant interfering sweeps, including by attempting to assess the physical scale of the interaction between two selective sweeps (Figure 3). Though limited by the selective interference that decreases the fixation probabilities at each locus, sweep interference may be more likely to happen than soft sweeps with recurrent mutations or trafficking, because it involves larger chromosomal regions, which increases the probability of occurrence of two beneficial mutations.

Beneficial substitutions may not be evenly distributed over time, but rather concentrated in short time periods following environmental changes, when a previously well-adapted population needs to climb a new adaptive peak (as, for instance, in Orr 1998). If so, the simultaneous occurrence of several beneficial mutations may not be unlikely, and interference of selective sweeps may alter to some extent our ability to detect positive selection in genome scans, adding a new confounding factor to demography (Jensen et al. 2005) or variable genomic features (mutation, recombination). Perhaps more readily, the search for interfering selective sweeps could be helpful in specific studies focusing on smaller candidate regions, in which several putative targets of selection have already been identified. In such cases, the analysis of the polymorphism pattern could provide information not only about the presence of selection, but also about the synchronicity of selective sweeps or the origin (migration vs. mutation) of beneficial alleles This could yield valuable insights into the adaptive history of a species (Camus-Kulandaivelu et al. 2008).

We assumed here that selective sweeps had independent (multiplicative) effects on fitness. Epistasis between loci contributing to adaptive traits has already been shown to generate linkage disequilibrium between those loci (Caicedo et al. 2004). Epistasis between selected loci may also influence the neutral polymorphism pattern of interfering sweeps in a specific manner, so that it could be possible to identify selective interactions a posteriori. For instance, a recent article revealed a double selective sweep at two closely linked chromosomal regions involved in the sex-ratio distortion of D. simulans (Derome et al. 2008). Though both regions are compulsory for meiotic drive to occur in the lab (Montchamp-Moreau et al. 2006), the functional relationship between these two regions is still questioned in natural populations (C. Montchamp-Moreau, personal communication). It may be possible to use the polymorphism pattern in this region to try to elucidate how those loci interact in natural populations. More work is needed to investigate if there can actually be a molecular signature of the interaction between loci.


We thank Frantz Depaulis as well as two anonymous reviewers for helpful comments on an earlier version of this manuscript. This work was supported by a Bourse de Docteur Ingénieur grant from the Centre National de la Recherche Scientifique (CNRS) to L.-M.C. F.H. and L.-M.C. are supported by grant ANR-06-BLAN-0128 from the CNRS.


Here, we want to describe the interactions between two selected loci and their neutral background. This can be tackled using the methodology of Barton and Turelli (1991) and Kirkpatrick et al. (2002), as done by Stephan et al. (2006). However, this is not fully necessary here, as the problem can be studied with a simple three-locus model: the two selected loci, plus a neutral locus. Different situations are investigated by changing the location of the neutral locus relative to the selected loci. In this appendix, we derive recursion for the gene frequencies at the three loci, the two- and three-locus linkage disequilibrium, and mean fitness.

For the sake of generality, let us consider three loci A, B, and C, located in that order on a chromosome, with r1 (resp. r2) the recombination rate between loci A and B (resp. B and C). The recombination rate between extreme loci A and C is then

equation M58

Assume that each locus has two alleles denoted by upper- and lowercase letters (A, a, B, b, C, and c). There are K = 8 possible gametic haplotypes:

equation M59

Let xk be the frequency of gamete haplotype k. We have

equation M60

and from basic genetic definitions, we can write expressions for the frequencies of the uppercase genotype at one, two, and three loci:

equation M61

The two-locus linkage disequilibrium between loci A and B writes

equation M62

The linkage disequilibria CAC and CBC for the two other pairs of loci are obtained seemingly by replacement, and finally the three-locus linkage disequilibrium is

equation M63

There are 36 possible diploid genotypes equation M64 to consider. The probabilities P(i, j, k) that a parent formed by the gametes i and j produces the gamete k after meiosis are given in Table A1. This table was derived using the Mathematica notebooks defined in Hospital et al. (1996).

Each locus may be selected or neutral depending on the case considered. This does not change the probabilities in Table A1, but simply the selection coefficient attributed to each locus.

The fitness of a diploid genotype composed of two gametic haplotypes i and j is

equation M65

where equation M66 is the number of copies of the favorable allele at selected locus sel1, and similarly for sel2, and where s1 and s2 are the corresponding selection coefficients. Note that in the text, we rather define the fitness (written with uppercase W) directly from equation M67 and equation M68 without reference to the haplotypes for the sake of clarity, so

equation M69

The frequencies x′ of the gametes at the next generation are obtained by

equation M70


equation M71


equation M72

where P(i, j, k) is taken from Table A1, w(i, j) is the fitness of diploid genotype (i, j) computed as in (A7), and equation M73 is the mean fitness in the population.

Now, we can use (A9) to compute the various quantities defined in (A4)–(A6) from one generation to the next. We use a prime (′) to denote the quantity at the next generation. It turns out that for all these quantities, with the notable exception of the three-locus linkage disequilibrium (A6), the expression does not depend on the respective positions of the loci (i.e., whether the neutral locus is “between” or “outside” the selected loci). Hence, without loss of generality we can write these quantities in terms of loci neu, sel1, and sel2 without taking account of their order on the chromosome. We get for the variation of gene frequency at any of the selected loci

equation M74

where i indicates the selected locus considered and j the other selected locus.

For the variation of gene frequency at the neutral locus we get

equation M75

And for the mean fitness,

equation M76

For the linkage disequilibrium at the next generation between the neutral locus and one selected locus, we get

equation M77

And for the linkage disequilibrium at the next generation between the two selected loci,

equation M78

Finally, the expression for the linkage disequilibrium at the next generation between three loci is too long to display. Instead, we give the frequency of the three-locus haplotype at the next generation,

equation M79


equation M80

i.e., equation M81 is the probability that there is no recombination either between the first and second locus or between the second and third locus on the chromosome. Note that equation M82 is the only term that depends on the positions of the loci on the chromosome, whereas all other terms depend only on the status of the loci (neutral or selected).

One can then plug (A17), as well as the recursion for two-locus linkage disequilibrium, and gene frequencies into (A6).


This appendix explains the calculation of the scaled delay time between the beneficial mutations at sel1 and sel2. This scaled time is used to account for the dynamics of the beneficial allele at sel1, which spends more time at low and high frequencies than at intermediate frequencies. Taking advantage of the fact that

equation M83

where epsilon is the beginning of the deterministic phase, the expected trajectory of the first beneficial mutation is

equation M84

which coincides with Stephan et al.'s (1992) Equation 3a. By reversing this equation, the expected time before reaching a frequency equation M85 is

equation M86

The threshold frequency of the beneficial mutation at sel1 when the beneficial mutation at sel2 is introduced is pt. We define the scaled delay time between the selected mutations as

equation M87

It is the proportion of the expected total duration of the deterministic phase reached by the selective sweep at sel1, when the beneficial mutation occurs at sel2. τ has meaning only for frequencies at which the dynamics of the selected locus are nearly deterministic; therefore τ = 0 corresponds to equation M88 and τ = 1 to equation M89 (epsilon was set to 40/20,000 on the basis of the observed course of the simulations). Note that the actual “simultaneous” case treated above (equation M90) is not strictly equivalent to τ = 0, as it includes the frequencies at which the dynamics are governed by the stochastic process.


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