Introduction

When an advantageous mutation arises and rapidly increases to high frequency under strong positive selection, neutral variants on the same chromosome (initially linked to the mutation) “hitchhike” along the mutation to high frequency. This rapid change in neutral allele frequencies generates a characteristic pattern of polymorphism, commonly referred to as a “selective sweep”. Meiotic recombination breaks the association between the advantageous and the neutral allele (the “hitchhiker”). Therefore, the pattern of a selective sweep is contained within a small map distance from the locus under selection. Signatures of selective sweeps include the local reduction of polymorphism (expected heterozygosity), skew of site frequency spectrum, and a unique spatial pattern of linkage disequilibrium. As vast amounts of genome-wide data becomes available, the characteristic patterns of genetic hitchhiking provide a powerful tool to identify candidate regions in the genome that were recently (or still are) under positive directional selection. Moreover, as the quality of genetic data improves, it might be possible to develop methods aiming to reconstruct the underlying evolutionary dynamics by “reverse engineering” hitchhiking patterns. This however requires to extend classical theory to situations that reflect organism-specific characteristics regarding particularities in e.g. the selective environment, demography, or mating structure.

[1] first provided a comprehensive mathematical analysis of this evolutionary process. Since then, remarkable advancements in the mathematical theory of selective sweeps were made [2]–[5]. Theories focused on the stochastic patterns of variation, mainly achieved through coalescent and diffusion approximations, in order to detect and interpret selective sweeps from DNA sequence polymorphism. Consequently, as more genomic data became available, clear cases of selective sweeps that confirm such theoretical predictions rapidly accumulated (reviewed in [6]–[8]).

Recent theoretical studies focus on the expansion of theory beyond the “standard model” of genetic hitchhiking. The standard model assumes that an advantageous mutation, arising as a single copy on a random chromosome, increases to high frequency under constant and homogeneous selective pressure in an ideal random-mating population of constant size. This model, the basic scenario of adaptive evolution that [1] considered, is simple enough to allow the application of diffusion and coalescent approximations and thus the prediction of stochastic patterns. However, a selective sweep in a real population must occur under a very complex demographic structure and under various modes of positive selection. Application of the standard model of genetic hitchhiking to the interpretation of actual data may thus lead to a serious problem. Recent studies addressed this problem by modeling selective sweeps that occur from standing genetic variation or recurrent beneficial mutations [9], [10], under arbitrary dominance of the beneficial allele [11], under selection on a quantitative trait [12], in newly derived populations [13]–[15], in geographically structured populations [16], and under the complex life cycle of malaria parasites [17], [18].

Homogeneity of selective pressure driving the beneficial mutation to a high frequency is an important assumption in the standard model of selective sweeps. Typically this is well justified even for a population that is distributed over multiple “patches” with different selective environments, if individuals move rapidly over different patches and also mate with other individuals from other patches. In that case, the population might be modeled to be panmictic under homogeneous selective pressure, which is given by the selective advantage of the beneficial allele averaged over all patches. However, as will be argued below, if mating is restricted to between individuals within the same patch, it can alter the effective rate of meiotic recombination and thus the strength of genetic hitchhiking. This may be important for many species in which mating occurs between individuals within a restricted range (under the same selective environment) but the young offspring (or seeds) are dispersed over a much wider range. Such species include plants that reproduce frequently by self-fertilization and animals that lay eggs in common breeding sites from which the young disperses into random habitats, or agents causing vector-borne parasitic diseases. Particularly plasmodium species, parasites that cause human malaria, are further important examples: male and female gametocytes produced inside a single human host enter a mosquito's gut during the blood meal and release gametocytes, which immediately fuse and undergo meiosis, producing sporozoites that are transferred to different hosts. Therefore, given that hosts constitute heterogeneous selective environments (“patches”) for parasites, an allele under selection experiences random switches of patches over malaria transmission cycles while mating always occurs between gametocytes from the same patch. This is an important consideration for malaria parasites in which strong patterns of selective sweeps due to the evolution of drug resistance were discovered [19]–[22].

This study investigates the hitchhiking effect of a mutant allele spreading over a heterogeneous environment, which is composed of patches with different selective pressures. Averaged over all patches, the mutant allele is advantageous over the wild type. This model of selection with random dispersal over patches between generations is known as the hard-selection Levene model (cf. [23], Ch. 6). [24] originally formulated this model assuming soft selection. While typically the Levene model is considered to study the maintenance of multiple alleles at the balance of selective pressures in different patches, we consider an overall advantage of the mutant allele that ensures the rapid increase of its frequency by directional selection. Our model also differs from the Levene model in which mating between individuals occur within patches.

After formulating the deterministic model and deriving the corresponding recursion equations, we study the effect of a single locus under positive directional selection on a neutral multialelic locus. We propose an analytical solution for the equilibrium frequencies and the expected heterozygosity at the neutral locus after the sweep is complete. We further derive approximations for the equilibrium heterozygosity that are easier to interpret. In particular we want to contrast within-patch mating and mating after random dispersion over the whole population. Even further, we present stochastic simulations in comparison to the analytic results of the deterministic model.

Methods

Results

Now we want to study genetic hitchhiking, i.e., the influence of selection at a single locus on a linked neutral locus. For this purpose we assume that the first locus is selected with two alleles and , and that the second locus is selectively neutral with finitely many alleles . We number the haplotypes such that stands for and stands for (). Moreover, we denote the recombination rate between the two loci by .

Dynamics at the Neutral Locus

Now we want to study the hitchhiking effect of the spread of an resistant allele at a single locus on neutral variation. As before denotes the frequency of the resistant allele . We have . Moreover, we denote the frequencies of the neutral allele with an -background and -background by , and , respectively.

In Analysis, subsection 2, we derive in generation to be(11)where(12)and

(13)(14)

In the last step we set for , , and . Hence, we defined patch as the breeding site, and is the proportion of the population mating in path .

Although, we could in principle derive analogously, we refrain from doing so. We are only interested in the case in which the allele sweeps through the population. Hence, at equilibrium vanishes, and all neutral alleles are linked to the allele . Hence, the equilibrium frequency of is given by . Deriving these frequencies allows to study genetic hitchhiking. In particular, we have(15a)where

(15b)From the above it is straightforward to calculate the equilibrium heterozygosity defined by(16)

The equilibrium heterozygosity depends on the initial allele-frequency distribution at the neutral locus, because does. However, as shown in Analysis (subsection 3) the relative expected heterozygosity defined by

is independent of the initial distribution of allele-frequency distribution. Here, E denotes the expectation (over the initial distribution of allele-frequencies), andis the initial heterozygosity. We summarize

Result 2.

The equilibrium frequency of the neutral allele is given by (15). The expected relative heterozygosity is calculated to be(17)where is defined in (15b).

Remark 2.

For the hard-selection Levene model, we need to set for all , which gives(18)which clearly is exactly the solution for standard hitchhiking.

The differences between our model and the hard-selection Levene model become obvious from the above remark. Whereas the dynamics at the selected model coincide for both models, differences occur at linked neutral loci. Not surprisingly, the hard-selection Levene model is equivalent to the standard haploid selection model. In particular, the relative heterozgosity which measures the hitchhiking effect (see section 3) does not coincide for the two models. Figures 1, 2, and 3 illustrate these differences.

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Figure 1. Heterozygosity as a function of .

Average relative heterozygosity (left y-axis) and (right y-axis) as a function of assuming two patches (). We assume either complete within-patch mating and dispersion (WD; ) according to the model introduced here, or the hard-selection Levene model (L; ). Solid lines correspond to exact solutions according to equations (15) and (18), respectively. Dashed lines show approximate solutions according to equation (25a) combined with equations (25b) and (25c), respectively. Dots represent the values obtained from stochastic simulations. Fitness values are shown in the boxes above the plot panels in (A) and (B). Stochastic simulations are based on repetitions for each parameter combination and . For the exact and approximate solutions we assumed to compensate for the deterministic solution's overestimation of heterozygosity due to the prolonged initial spread of the beneficial mutation in the deterministic model.

https://doi.org/10.1371/journal.pone.0061742.g001

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Figure 2. Heterozygosity as a function of .

Average relative heterozygosity as a function of . See legend of Figure 1 for more details.

https://doi.org/10.1371/journal.pone.0061742.g002

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Figure 3. Exact vs. approximate average relative heterozygosity.

Average relative heterozygosity as a function of as given by (15) and (18), and equation (25a) combined with equation (25b). Two patches with were assumed. Moreover, fitness parameters and initial frequencies are shown in the boxes above the plot panels in (A), (B), (C), and (D).

https://doi.org/10.1371/journal.pone.0061742.g003

The analytic solution (17) is insofar not satisfying as it is iterative and difficult to interpret. We will therefore derive approximations that have a simpler form and are easier to interpret in terms of the involved parameters.

Discussion

While adaptive evolution in reality follows complex patterns (demography, heterogeneous selection pressures, spatial structure, mating behavior, etc.), such processes can often be accurately described within the idealized framework formed by standard population-genetic assumptions (constant homogeneous selection pressures, constant population size, random mating). Deviations from standard assumptions - particularly heterogeneities in selective pressures - are obviously important in allopatry and parapatry. However, even individuals living in sympatry might experience substantial differences in selective pressures. Examples include selection for herbicide resistance in weeds [26]–[28], stress tolerance in insects and weeds in agriculture, insecticide resistance in bed bugs [29]–[32], drug resistance in vector borne diseases (see below). Whereas in these examples candidate regions under selection might be inferred with population-genetic methods that build up on standard theory, substantial errors could result when attempting to reconstruct the underlying evolutionary dynamics (e.g., estimating selection coefficients, speed of evolution, recombination rates, etc.) from the selective sweep patterns. To avoid misinferences under such scenarios, it is therefore necessary to validate the applicability of standard population-genetic theory, and - if appropriate - adapt existing theory, particularly since many of the mentioned examples are matters of economic relevance and/or global health interest.

For instance, Plasmodium parasites causing human malaria typically experience different ‘environmental conditions’ depending on characteristics of human hosts determining selective regimes (drug treatment, drug dosage, immune response, levels of host-acquired or natural immunity, etc.). Parasites conferring resistance to antimalarial drugs are advantageous only in hosts treated with the respective drugs, whereas they are slightly deleterious in untreated hosts due to metabolic costs. In parallel, sexual reproduction occurs inside the mosquito vector, randomly but exclusively between parasites that were extracted from the same host, manifesting another deviation from standard assumption. Heterogeneous selection pressures act also on a spatial scale because drug-deployment policies and control interventions are country specific. This is particularly relevant along the borders of Cambodia, Laos, Myanmar, and Thailand where the containment of emerging artemisinin resistance is of fundamental importance to sustain successful malaria control [33]. Inferences based on standard population-genetic assumptions might be misleading as parasites experience highly varying selective environments and severe inbreeding is immanent to the specifics of malaria transmission.

More generally, parasites or pathogens that sexually reproduce within hosts might experience radically heterogeneous selection pressures, as immune responses may occur differently across organs or within specific tissues. Sexual reproduction might be common even in fungal pathogens [34]. In agriculture patches of contrasting selective regimes are created in sympatry by human interventions (cf. [35], [36]). The use of fertilizer, manure, herbicides, pesticides along with interventions such as plowing and irrigation varies across farmed land. Therefore, insects or weeds might experience radically different selective conditions across nearby acres. A striking example of a rapid evolutionary change under such a setting is the fast progression of glyphosate (“roundup”) resistance in many species of weeds, economically challenging US agriculture. Genetic understanding of glyphosate resistance will require the detection and analyses of selective sweeps in the plants, including those reproducing by self-pollination and long-distance seed dispersal.

In this study we introduced a model for heterogeneous selection in sympatry within a haploid population that randomly disperses across patches in every generation. Viability selection acts differently within the patches and mating occurs randomly within or between patches. In the limiting case that mating occurs randomly between all patches, the model reduced to the hard-selection Levene model (cf. [23], Ch. 6), which is identical to the standard selection model. However, if mating occurs exclusively within demes, the deviations from the standard model can be substantial. We showed that the dynamics at a single selected locus are independent of the dispersal pattern. Namely, they are solely determined by the average selection intensities across patches. However, as soon as two or more linked loci are considered deviations from standard-population-genetic assumptions become apparent. Particularly, we studied how the genetic variation at a neutral locus is affected as a beneficial mutation sweeps at a nearby linked locus.

We were able to derive an analytic solution for the allele frequencies at a neutral locus after the beneficial mutation became fixed. As the analytic solution is complicated we also derived approximations, which allow for clear and simple interpretations. Namely they reflect the frequency change driven by the selective pressure averaged over patches, however adjusted by a factor determining the relative importance of the patches. As long as differences in selection pressures are moderate the hitchhiking effect is accurately described by standard population-genetic theory. However, if selection pressures are extreme as it might be the case in the above mentioned examples, heterogeneities in selection pressures in combination with intra-patch mating leads to stronger reductions in genetic variation than predicted by the standard model. The reason is as follows. Radically reversing directional selection across patches leads to mating only between individuals carrying the allele that allows survival within the respective selective environments, thus greatly increasing the effect of inbreeding. Hence, meiotic recombination is less efficient to restore genetic variation. This effect however cannot be just summarized by an adjustment of the recombination rate. In fact the unique mating scheme leads to a process for which selection and recombination cannot be decoupled.

We also performed stochastic simulations to verify the results of the deterministic model's analytic prediction. As expected the deterministic solutions were underestimating the reduction of genetic variation at neutral loci. However, as usual this can be compensated by adjusting the effective initial frequency of the advantageous allele, which reflects the shorter allele frequency trajectory of the advantageous allele conditional on its escape from extinction by random genetic drift.

In general our results are informative to properly interpret selection coefficients when these are attempted to be measures from the patterns of selective sweeps. Unfortunately, appropriate data is unavailable for the mentioned examples to which our model would apply (pesticide and herbicide resistance). Nevertheless, as the examples are of great economic interest, and as population genetic theory continues to advance such data hopefully become available soon. Anyhow, the model is applicable to malaria where attempts have been made to link estimates of selection to the hitchhiking pattering (e.g. [19], [22]).

The hitchhiking effect revealed in this study might be compared to that of another study assuming the subdivision of population into many small demes or patches [16], [37]. They predicted the reduced strength of hitchhiking (higher heterozygosity), in contrast to our current result, due to population subdivision. Their model however assumes homogeneous selective pressure over demes and limited migration of individuals between demes. In such a case, the delay in the propagation of advantageous allele into the entire population provides more opportunities for recombination that breaks the hitchhiking. Most populations in nature would violate the assumptions of both studies (instantaneous dispersal among demes of the current study and homogeneous selective pressure in [16]). Further investigation is needed for the joint effect of the two forces.

Analysis

Acknowledgments

The authors gratefully acknowledge the work of Hua Chen and an anonymous reviewer as well as their helpful comments on an earlier draft of this manuscript.