Here, I study diversity at 91 X-linked coding regions in D. miranda (~1.1 kb on average), in a sample of 14 individuals. Table 1 gives an overview of the polymorphism summaries across the regions investigated, and their level of divergence to D. pseudoobscura. For locus-specific estimates of polymorphism statistics and divergence, see Additional file 1. A total of 489 synonymous and 144 replacement polymorphisms were observed. Average pairwise diversity is 0.62% at synonymous site, and roughly 15-fold less at replacement sites (0.04%, see Table 1). Similarly, divergence is lower at replacement sites compared to synonymous sites (Table 1). There is a general skew in the allele frequency spectrum towards rare variants, as measured by Tajima's D 26, both at synonymous sites and at replacement sites (Table 1). However, replacement sites are skewed more strongly towards rare variants compared to synonymous sites (D = -0.80 vs. D = -0.44, Table 1). Reduced diversity, reduced divergence and more low-frequency variants are all expected if amino-acid sites are under stronger purifying selection than synonymous sites 27. The skew of synonymous sites could reflect non-equilibrium demography, or positive selection at linked sites (see below).

Here, I use synonymous sites as a neutral marker in MK tests, and to detect hitchhiking effects associated with adaptive protein evolution. In Drosophila, synonymous codons are not used randomly, but instead some codons are used preferentially over a different codon encoding for the same amino acid 2829. Thus, synonymous sites are not necessarily free of selective constraints. Nevertheless, selection for codon bias is strongly reduced in D. miranda, and synonymous sites are evolving probably close to neutral in this species 2530. Thus, synonymous polymorphisms are a suitable "almost neutral" marker for tracking adaptive events in D. miranda.

The MK test and its extensions were used to test for adaptive protein evolution in D. miranda (Table 2). The fraction of amino-acid substitutions that have been fixed by positive selection (α) is estimated using three slightly different approaches 1231. Segregating slightly deleterious amino-acid mutations will bias the estimate of α downwards, because slightly deleterious mutations tend to contribute relatively more to polymorphism than they do to divergence, when compared to neutral mutations 23233. Slightly deleterious mutations segregate at lower frequencies than neutral alleles, and consistent with the idea of segregating deleterious amino-acid mutations, replacement polymorphisms do segregate at lower frequencies than synonymous mutations in D. miranda (Table 1). I attempt to reduce the effect of slightly deleterious mutations in my analysis, by also removing polymorphisms at a frequency of 10% or lower, at both synonymous and replacement sites (Table 2). If all polymorphisms are considered, estimates of α range from 12%–41%, depending on the estimation method used (Table 3). If low-frequency variants are ignored, I estimate α to be between 44%–61% (and the 95% confidence intervals do not overlap zero, Table 3). Thus, roughly half of the protein divergence between D. miranda and D. pseudoobscura is driven by positive selection. These numbers are in close agreement to estimates of α obtained in D. melanogaster and D. simulans if similar approaches are used to estimate α 12391431; but also see 4.

Average synonymous divergence between D. miranda and D. pseudoobscura is low, roughly only 4% (Table 1). It has been suggested that low levels of divergence can lead to upwardly biased estimates of α for two reasons 831. First, advantageous mutations spread through a population much more rapidly than neutral mutations. Thus, the divergence time may have been to short for a stochastic steady state for fixations to have been reached, resulting in an upward bias in the estimation of α 31. The divergence time between D. miranda and D. pseudoobscura is about 7-fold larger than the population coalescence time, very similar to that observed for D. melanogaster and D. simulans 934. Thus, if low divergence leads to a bias in the estimation of α, a similar bias may be expected in the two species comparisons. Secondly, since only a single D. pseudoobscura sequence is used for outgroup comparison, some of the inferred divergence includes undetected segregating polymorphism, which could also result in an upward bias in the estimation of α 8. To avoid potential biases in inferring the fraction of beneficial amino-acid fixations due to low levels of divergence, I also estimated α using divergence to a more distantly related species. To this end, D. affinis was employed, a species which shows about 25% synonymous divergence to D. miranda at synonymous sites (see Table 1). Interestingly, using D. affinis as an outgroup species results in similar, although slightly lower, estimates of α (i.e. 30%–40%, see Table 3). If α is estimated along the D. miranda lineage, using a reconstructed D. miranda – D. pseudoobscura ancestral sequence, estimates of α are increased (Table 3). Thus, while α may be overestimated for short divergence times, D. pseudoobscura appears sufficiently diverged from D. miranda to result in little bias in estimating α due to low levels of sequence divergence.

Positively selected amino-acid mutations should leave characteristic signatures at surrounding genomic regions. One such signature is reduced neutral diversity surrounding sites that are targets of positive selection; i.e. selective sweeps 35363738. Indeed, several recent studies have found reduced levels of synonymous diversity at genes with high rates of protein evolution in D. melanogaster 1739 and D. simulans 15.

Levels of synonymous diversity (π_s) are reduced in protein-coding genes with high rates of amino-acid evolution (K_a) in D. miranda (R = -0.265, p < = 0.01, rank correlation test; Figure 1), as expected if fast evolving proteins undergo more frequent adaptive evolution. I find no correlation between rates of substitutions at synonymous sites (K_s) and K_a (R = 0.0809, p < = 0.44, rank correlation test), suggesting little mutation rate variation among the regions studied.

The recurrent hitchhiking model predicts a stronger skew towards low-frequency variants at faster evolving genes 3840. In accordance with this expectation, I find a significant negative correlation between K_a and Tajima's D (R = -0.2077, p < = 0.05, rank correlation test, Figure 2A). Recurrent hitchhiking is also predicted to reduce the efficacy of natural selection against very weakly selected sites, such as synonymous sites experiencing codon bias selection 4142. As a measure for codon bias selection experienced by each gene, I calculate the frequency of optimal codons (Fop), as identified in D. pseudoobscura 30. Figure 2B shows that Fop is negatively correlated with rates of protein evolution (R = -0.3523, p < = 0.0008, rank correlation test), i.e. genes undergoing more adaptive evolution generally show less biased codon usage. This is consistent with the notion that adaptive protein evolution interferes with selection for codon usage at linked sites (but see also 434445).

Reduced diversity at synonymous sites, a skew toward low-frequency variants, and reduced selection for codon usage bias in fast evolving proteins are all features expected under recurrent adaptive amino-acid evolution. I fit a recurrent hitchhiking model to the relationship between synonymous site diversity (π_s) and amino acid divergence (K_a), to quantify selection parameters for beneficial mutations 13. The diversity reduction under the hitchhiking model employed 13 depends on the rate at which beneficial mutations occur (λ), and their average selection coefficients (s). Assuming adaptive protein evolution solely accounts for the observed diversity reduction at fast evolving genes, and no heterogeneity in α among genes, as suggested in Drosophila 831, λ can be directly related to α, the fraction of adaptive protein divergence (λ = αK_a/2T, where T is the species divergence time, see Methods). In Figure 1, I show the least squares fit of the D. miranda polymorphism data to this recurrent hitchhiking model. For comparison, I performed the same fitting procedure using a data set of 137 X-linked protein-coding genes from a sample of 12 D. melanogaster alleles sampled from a Zimbabwe population, which also shows a correlation between K_a and π_s 17. Table 4 gives an overview of relevant population summary statistics for D. miranda and D. melanogaster and estimated selection parameters for beneficial amino-acid mutations. Note that the positive selection parameters inferred for D. melanogaster in this study differ slightly from those presented in ref. 17, where the same data set using a simulation-based method was analyzed. However, while the exact absolute values of selection parameters depend on the methodology used, it is the comparison between D. miranda and D. melanogaster that is the main focus of this paper. I estimate the neutral population mutation rate (i.e. in the absence of genetic hitchhiking effects) to be θ = 0.99% per site in D. miranda and θ = 2.81% in D. melanogaster (Table 4). The model also yields an estimate of the product αs = 1.2 × 10^-3 in D. miranda and αs = 1.2 × 10^-5 in D. melanogaster.

In the recurrent hitchhiking model used, s and α (or λ) are conflated parameters and only their product can be estimated. However, I can infer s using an independent estimate of α. I estimate α = 44% for D. miranda and α = 51% for D. melanogaster, using a maximum likelihood method 31. This value of α implies that s ~ 2.7 × 10^-3for D. miranda and s ~ 2.3 × 10^-5 for D. melanogaster. Given estimates of λ, N_e and s, we can also estimate the rate at which beneficial mutations arise per site and generation (u_ben = λ/(4N_e s), see Methods). The beneficial mutation rate is u_ben ~ 6.5 × 10^-14 in D. miranda and u_ben ~ 4.7 × 10^-12 in D. melanogaster. Thus, while the inferred rate of adaptation (i.e. the fixation rate of beneficial mutations) is similar between D. miranda and D. melanogaster (λ ~ 4.0 × 10^-10 in D. miranda and λ ~ 7.0 × 10^-10 in D. melanogaster, corresponding to λ 2N_es ~ 1.2 × 10^-6 and λ 2N_es ~ 5.1 × 10^-8, Table 4), I estimate a lower beneficial mutation rate in D. miranda but larger average effects of beneficial amino-acid mutations.

If the rate of beneficial mutations and their selective effects are constant across species, larger populations are expected to show higher rates of adaptation, both because they generate more mutations and selection is effective on a greater proportion of mutations 1846. Kimura and Ohta 47 noticed that rates of protein evolution are rather similar across taxa with different population sizes, and argued against models of adaptive substitutions to explain protein evolution. Here, I compare rates of protein evolution in two Drosophila species that differ about 5-fold in their levels of neutral variability, but find little difference in rates of adaptive protein divergence between them.

Specifically, two independent types of evidence suggest that D. melanogaster and D. miranda both show high rates of adaptive protein evolution. First, application of the MK test suggest that a similar fraction of amino-acid mutations is driven to fixation in both species groups, and my estimate of λ, the rate of selective sweeps, is very similar between the two species. Second, both species show a correlation between rates of protein evolution and synonymous site diversity, as expected under recurrent adaptive protein evolution 1739. Fitting a recurrent hitchhiking model to polymorphism data suggests a higher value of αs in D. miranda than in D. melanogaster. Since α is estimated to be very similar between species, s is inferred to be larger in D. miranda.

What would we expect α, the rate of adaptive protein divergence, to be in D. miranda using selection parameters inferred from D. melanogaster but accounting for differences in their synonymous site diversity? The rate of adaptive divergence K_ben can be calculated as K_ben = 2N*u_ben *2s*2T. Using estimates of u_ben and s from D. melanogaster (Table 4), we would expect the rate of adaptive amino-acid divergence in D. miranda to be 0.107% per site. This means that only 18% of total amino acid divergence was driven by positive selection, i.e. α = 0.18. This value of α lies below the confidence limits we estimate for α in D. miranda, using MK approaches (Table 3). Thus, assuming that selection parameters are similar between the species, D. miranda displays higher rates of protein evolution than would be expected given its level of diversity. Below, I discuss several implications of this finding and possible explanations.

Levels of synonymous diversity are lower in D. miranda relative to D. melanogaster (see Table 4), suggesting a smaller effective population size in D. miranda. However, levels of diversity only contain information about the recent effective population size of a species, and it is possible that both species had similarly sized populations for much of their evolutionary history. Indeed, there is some evidence that D. miranda has experienced a reduction in its effective population size given reduced levels of diversity compared to its closest relatives, reduced selection to maintain codon bias, or mulitlocus patterns of diversity 222330. Thus, much of the adaptive evolution detected in D. miranda could reflect selection in a larger, ancestral population, or selection in D. pseudoobscura (which based on levels of synonymous polymorphism, is thought to have a larger population size 22) since divergence also includes fixations along the D. pseudoobscura lineage. However, similar estimates of α are inferred, regardless of whether D. pseudoobscura or D. affinis is used as the outgroup species, or whether the reconstructed ancestral sequence of D. miranda and D. pseudoobscura is used (Table 3). While estimates of α might be biased upwards if divergence between species is low 831, a large fraction of amino-acid mutations along the D. miranda are clearly driven by adaptive evolution. Furthermore, if D. miranda has experienced a recent reduction in its effective population size, the estimate of α is likely to be biased downwards, due to segregating deleterious mutations (see below). Similar arguments of a reduced population size in D. melanogaster compared to its close relative to D. simulans have been invoked to account for lower levels of diversity at autosomal regions, decreased selection for codon bias and patterns of polymorphisms in D. melanogaster 484950. However, X-linked diversity appears similar between the two species 934, and estimates of adaptive protein evolution are comparable 8931. Thus, the exact dynamics of the long-term effective population size is unclear in both species, but both may have undergone a recent contraction in size.

While inferences of adaptive evolution based on the MK test are sensitive to assumptions about the long-term effective population size of a species, the correlation between synonymous diversity and rate of protein evolution provides an independent conformation of high rates of adaptive protein evolution in both species. This observation is less affected by fluctuations in the long-term effective population size, since reduced diversity in fast evolving proteins reflects the action of very recent selection (i.e. in the order of the population coalescence time). Interestingly, the rate of selective sweeps inferred from the observed correlation between π_s and K_a is very similar for the two species. Thus, despite uncertainties about the long-term effective population size in both D. melanogaster and D. miranda, several independent lines of evidence suggest similar and high rates of protein evolution in both species. It will be of great interest to study a diverse set of species that differ in their inferred effective population sizes, to better study the influence of population size on rates of adaptive evolution.

The expectation of less adaptation in smaller populations relies on the assumption that the mutation rate and the strength of selection for beneficial mutations are constant among species investigated. However, I infer that both the beneficial mutation rate and the strength of selection are different between D. melanogaster and D. miranda. But how plausible is such a scenario? In a species with a larger effective population size, a smaller fraction of mutations is effectively neutral, and one might expect such a species to be better adapted since more advantageous mutations will fix 18. This could change the distribution of fitness effects because, as a species adapts, its fitness is expected to move closer to an optimum 5152. Thus, one could intuit that better adapted species have fitness distributions where beneficial mutations are weaker on average, but it may be harder to explain why the rate of adaptive mutations should be higher in that case.

There is some experimental evidence that the distribution of fitness effects is different between species that have different effective population sizes. Silander et al. 53 performed a mutation-accumulation experiment where bacteriophage populations were passaged through different population sizes. As expected, the small population size lines had lower fitness because they had accumulated more deleterious mutations. Surprisingly, about 15% of the mutations in the small population size lines were adaptive, in contrast to almost none in the large population size lines. The mean effect of mutations, however, did not seem to differ between lines. In comparisons between D. miranda and D. melanogaster, however, I infer that mutation rates are actually lower in the species with the lower population size, while fitness effects for beneficial mutations appear much larger.

It is also possible that the distribution of beneficial amino-acid mutations is conserved across species, but that biases in estimating α result in a substantial underestimation of the rate of adaptive amino-acid divergence in D. melanogaster. A schematic model of this effect is shown in Figure 3. Assume the rate of mutation and the strength of selection for adaptive amino-acid changes are constant between species. Because of its larger population size, fewer advantageous mutations will be effectively neutral in D. melanogaster and many adaptive substitutions of weak effect can fix in this species. In contrast, many more mutations are effectively neutral in D. miranda, but the average effect of the mutations incorporated would be larger. This would cause us to infer a larger beneficial mutation rate, but smaller selection coefficients in D. melanogaster, as observed.

While this model could account for the observed differences in rates and effects of beneficial amino-acid mutations, it would also predict that D. melanogaster shows much higher rates of adaptive protein evolution than D. miranda. However, my estimate of α is very similar between the two species. This discrepancy could potentially be explained by a bias in estimating α from MK tables, which assumes that all amino-acid polymorphism segregating in the population are neutral 7. However, if the fraction of beneficial amino-acid mutations is high and selection is weak, a substantial proportion of the amino-acid polymorphism segregating in the population might actually represent beneficial mutations. For example, Sawyer et al. 4 estimate that about half of all amino acids segregating in a population sample from D. simulans are beneficial. Including these mutations into MK tables will downward bias our estimate of α. Thus, if only very few amino-acid mutations are neutral in D. melanogaster (i.e. the majority is either deleterious or slightly beneficial), then a large fraction of amino-acid mutations that segregate in the population will be beneficial (once removing the low-frequency class), resulting in an underestimate of α using the MK framework employed here. In contrast, a larger fraction of amino-acid mutations will be effectively neutral in D. miranda, due to its smaller current population size, and thus a much smaller proportion of the observed amino acid polymorphisms are beneficial. This will cause the estimate of α to be much closer to its true value in D. miranda, while α might be substantially underestimated in D. melanogaster.

Consistent with a substantial fraction of amino-acid mutations segregating in D. melanogaster being beneficial, amino-acid polymorphism is less reduced relative to synonymous sites in D. melanogaster compared to D. miranda (π_a/π_s = 0.08 in D. melanogaster vs. π_a/π_s = 0.07 in D. miranda). Purifying selection against deleterious amino-acid mutations would be more effective in a larger species, predicting the opposite patterns, while the π_a/π_s ratio should be similar among species if amino-acid polymorphisms are neutral. In fact, if half the amino-acid polymorphisms are removed from D. melanogaster (i.e. they are assumed to be beneficial instead of neutral), the maximum likelihood estimate of α becomes closer to 0.8 in that species. In this case, we would conclude that D. miranda shows reduced rates of adaptation, in accordance with expectations based on its smaller current effective population size. While this model could qualitatively account for the observed patterns with regards to population size and adaptive evolution, it remains to be seen whether it can also quantitatively do so.

In the above considerations I assume that both the excess divergence at amino acid mutations relative to polymorphisms as well as the correlation between neutral diversity and protein evolution are entirely driven by beneficial mutations. However, for both interpretations, weakly deleterious mutation models pose a potential problem.

Slightly deleterious nonsynonymous mutations can yield biased estimates of α using MK methods, and the direction of this bias depends on the demography of the population. As mentioned before, if the population size has been relatively stable, the estimate of α is likely an underestimate, because slightly deleterious amino-acid mutations tend to contribute relatively more to polymorphism than they do to divergence when compared with neutral mutations. However, slightly deleterious mutations can lead to an overestimate of α if population sizes have expanded, because mutations that might have been fixed in the past, when the population size was small, no longer segregate as polymorphisms. Even fairly modest increases in population size can create artifactual evidence of adaptive evolution 33. Thus, it is possible that D. miranda has undergone a recent population size expansion (i.e. its current N_e is larger than the ancestral one) and that α is therefore overestimated. However, as mentioned above, if anything D. miranda appears to have gone through a population size decrease based on several consistent patterns of polymorphisms and divergence 222330. This suggests that α has not been overestimated in D. miranda because of a population size increase.

I invoke recurrent hitchhiking to explain the observed correlation between K_a and π_s in D. miranda; recurrent beneficial amino acid mutations reduce variation more frequently at faster evolving genes, causing a reduction in codon bias and an excess of low frequency variants. It is, however, formally also possible to explain the observed correlation between K_a and π_s solely by deleterious mutations models. The removal of recurrent deleterious mutations from the population reduces the local effective population size, causing regions with higher deleterious mutation rates (or lower recombination rates) to harbour reduced levels of neutral diversity; i.e. background selection 5455. A reduction in N_e, in turn, predicts an acceleration in the rate of accumulation of very weakly deleterious mutations 3256, including possible unpreferred codons or slightly deleterious amino acid mutations. Weak background selection can also distort the allele-frequency spectrum of neutral mutations towards rare variants 5657. Thus, selection against weakly deleterious mutations can by itself account for reduced levels of neutral diversity associated with increased rates of accumulation of slightly deleterious amino-acid mutations 3256, without invoking any beneficial mutations.

In D. miranda, the correlation between K_a and π_s is stronger than in D. melanogaster. This would suggest a larger fraction of slightly deleterious amino-acid mutations in the roam of weak background selection at some genes in D. miranda, causing a stronger reduction in N_e and π_s and higher rates of accumulation of deleterious amino-acid mutations. Again, a distribution of fitness effects against deleterious amino-acid mutations may in principle explain the pattern, if a larger fraction of deleterious amino-acid mutations in D. melanogaster is selected against efficiently. That is, only few amino-acid mutations with a relatively small range of negative selection coefficients cause weak background selection in D. melanogaster, while the majority of deleterious amino-acid mutations are selected against effectively. In contrast, more slightly deleterious amino-acid mutations segregate in D. miranda, resulting in a larger variance in the amount of weak background selection among genes. It is also possible that there is more heterogeneity in rates of recombination among the loci sampled in D. miranda, since physical and genetic map positions are not known for the loci investigated. Recombination rate variation may result in heterogeneous levels of background selection along a chromosome, and thus cause the local N_e (and the efficacy of selection) to vary among regions. Indeed, there is evidence for recombination rate heterogeneity in D. pseudoobscura, where mild "recombination hotspots" were discovered 58. Unfortunately, no genetic map exists in D. miranda, preventing us from directly investigating the effect of recombination rate heterogeneity on levels of polymorphisms in this species. However, as Andolfatto 17 points out, there are several empirical arguments against weak background selection causing the observed correlation between K_a and π_s, such as no general elevation of K_a in regions of the Drosophila genome with reduced recombination rates 4259, and no effect of gene density on rates of protein evolution 17.

In a series of papers, Gillespie examines the role of population size in population genetical models of molecular evolution, using extensive computer simulations 6061. Gillespie has defined three domains based on their rates of substitutions, which he terms the Ohta domain (the rate of substitution decreases with increasing population size), the Kimura domain (the rate of substitution remains close to the mutation rate) and the Darwin domain (the rate of substitution increases with increasing population size). Not surprisingly, he finds that nearly neutral deleterious mutation models all fall within the Ohta domain where the rate of evolution is inversely proportional to population size. He also finds that the normal-shift model (a model of positive selection) appropriately falls within the Darwin domain, but that the rate of substitution does not linearly increase with populations size and is substantially reduced relative to the expectation under the independence among sites model. Surprisingly, Gillespie also finds that the fluctuating selection, neutral, and overdominance model all lead to the Kimura domain, where the rate of molecular evolution is independent of the population size. In addition, adaptive evolution can actually cause the rate of substitution of deleterious alleles at a linked locus to increase with increasing population size 6062. Thus, the associations between N_e and rates of evolution might be complex, and intuitions drawn from very simple models of positive and negative selection might not apply to natural populations. Future work will be needed to assess how more complicated selection models, including selection from standing variation or mutation selection balance 6364, and interactions between beneficial and deleterious mutations will influence rates of molecular evolution and variation.

A total of 91 X-linked coding regions were surveyed in this study with a sample size of 14 D. miranda alleles. The following strains of D. miranda were used in the analysis, with their population origin given in parenthesis: 0101.3, 0101.4, 0101.5, 0101.7 (Port Coquitlam, BC, Canada); 0101.9, MA28, MA32, MA03.1, MA03.2, MA03.3, MA03.4, MA03.5, MA03.6 (Mather, CA, USA); SP138, SP235, SP295 (Spray, OR, USA); MSH22, MSH38 (Mt. St. Helena, CA, USA).

The D. pseudoobscura genome sequence was used to provide estimates of divergence 65. For a subset of 51 loci, the orthologous D. affinis allele was surveyed (strain 14012-0141.01 or 14012-0141.02). All genes were selected randomly with respect to gene function. Information about the specific loci surveyed and primers used can be found in Additional File 1.

Each ~1200 base pair region was PCR-amplified from genomic DNA extracted from single male flies, and primers and nucleotides were removed using Exonuclease I and Shrimp Alkaline Phosphatase. Cleaned products were sequenced on both strands with the original PCR primers and internal sequencing primers, using Big-Dye (Version 3, Applied Biosystems) and run on an ABI 3730 capillary sequencer. Sequence traces were edited using Sequencher (Gene Codes) software and multiple sequence alignments were generated using MUSCLE http://www.drive5.com/muscle/ with protein-alignment-assisted adjustments to preserve reading frames. Sequences have been deposited in Genbank under accession numbers (XXXX-XXXX).

For comparison with D. melanogaster, I used a published dataset of 12 individuals sequenced from a Zimbabwe, Africa population and their divergence to D. simulans 17.

The estimated number of synonymous sites, nonsynonymous sites, average pairwise diversity (π), average pairwise divergence to D. pseudoobscura (D_xy), as well as counts of the number of polymorphisms (S) and the summary of the frequency distribution of polymorphism frequencies, Tajima's D (Tajima 1989), were calculated using a library of Perl scripts ("Polymorphorama") written by the author and P. Andolfatto. The number of nonsynonymous and synonymous sites were estimated using the method of 66. Average pairwise diversity (π) and divergence (D_xy) estimates were corrected for multiple hits using a Jukes-Cantor correction 67. Multiply hit sites were included in all analyses but insertion-deletion polymorphisms and polymorphic sites overlapping alignment gaps were excluded. For lineage-specific estimates of divergence, I reconstructed a D. miranda – D. pseudoobscura ancestor sequence (ANC) by maximum likelihood, using D. affinis as an outgroup, as implemented in the codeml program of PAML. Locus-specific estimates of levels of diversity and divergence can be found in Additional File 1. To estimate codon bias selection at each gene, the frequency of optimal codons (as inferred from D. pseudoobscura, see 30) is used.

The fraction of amino acid mutations driven by positive selection, α, is estimated using three slightly different approaches based on the MK test 7, as implemented in the DoFEv2 software package (kindly provided by A. Eyre-Walker). The number of divergent sites were corrected for multiple hits using a Jukes-Cantor correction. I also exclude singleton mutations, to minimize the downward bias in inferring α due to segregating deleterious amino acid mutations.

Following 17, I use the relationship between synonymous site diversity (π_s) and amino acid divergence (K_a) to quantify recurrent selection parameters. I use the analytical approximation of 13, to jointly estimate the strength of selection (s) and the rate of adaptive substitution per site per generation (λ). The expected nucleotide diversity at neutral sites is

E ( π ) = θ ρ ρ + k γ λ , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyrauKaeiikaGIaeqiWdaNaeiykaKIaeyypa0tcfa4aaSaaaeaacqaH4oqCcqaHbpGCaeaacqaHbpGCcqGHRaWkcqWGRbWAcqaHZoWzcqaH7oaBaaGccqGGSaalaaa@3E05@

where θ is the population mutation rate, ρ is the recombination rate per site per generation, k is a constant ≅ 0.075, and γ (= 2N_es) is the intensity of positive selection (where N_e is the effective population size of the species and s is the strength of selection). Like 17, I assume that the rate of selective sweeps in the neighborhood of a focal neutral site is determined by the local rate of amino acid substitution at a gene K_a, and that some constant fraction of amino acid divergence at each locus, α, was driven to fixation by positive selection. Therefore, the rate of selective sweeps due to recurrent adaptive amino acid substitutions at a locus is λ = αK_a/2T, where T is the divergence time in generations between D. miranda and D. pseudoobscura. Equation (1) can be re-written as

E ( π ) = θ ρ ρ + k 2 N e s ( α K a / T ) , MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyrauKaeiikaGIaeqiWdaNaeiykaKIaeyypa0tcfa4aaSaaaeaacqaH4oqCcqaHbpGCaeaacqaHbpGCcqGHRaWkcqWGRbWAcqaIYaGmcqWGobGtdaWgaaqaaiabdwgaLbqabaGaem4CamNaeiikaGIaeqySdeMaem4saS0aaSbaaeaacqWGHbqyaeqaaiabc+caViabdsfaujabcMcaPaaakiabcYcaSaaa@4797@

and equation 2 allows levels of neutral diversity to be directly related to the extent of adaptive amino acid divergence at each locus (i.e. α K_a).

Average recombination rates in D. pseudoobscura are higher than those observed in D. melanogaster; average recombination rates across the X chromosome in D. melanogaster equal 3.3 cM/Mb 68, while the average recombination rate across the X chromosome in D. pseudoobscura is 8.3 cM/Mb 69. Estimating T=(K¯s−θ)/2μ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemivaqLaeyypa0ZaaSGbaeaacqGGOaakcuWGlbWsgaqeamaaBaaaleaacqWGZbWCaeqaaOGaeyOeI0IaeqiUdeNaeiykaKcabaGaeGOmaiJaeqiVd0gaaaaa@37FB@ and N_e = θ/3μ require an estimate of the neutral mutation rate, μ. I assume that μ^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqiVd0MbaKaaaaa@2D9B@ = 5.8 × 10^-9 per generation, the estimated average rate for single nucleotide mutations from D. melanogaster mutation-accumulation lines 70. With these parameter estimated and π_s and K_a for 92 loci, a least-squares method is used to find the values of θ and αs that minimize the sum of the squared deviations between observed π_s and E(π) predicted by the model, using the R statistical package.