THE RELATION BETWEEN THE NEUTRALITY INDEX FOR MITOCHONDRIAL GENES AND THE DISTRIBUTION OF MUTATIONAL EFFECTS ON FITNESS

Authors


Abstract

We explore factors affecting patterns of polymorphism and divergence (as captured by the neutrality index) at mammalian mitochondrial loci. To do this, we develop a population genetic model that incorporates a fraction of neutral amino acid sites, mutational bias, and a probability distribution of selection coefficients against new nonsynonymous mutations. We confirm, by reanalyzing publicly available datasets, that the mitochondrial cyt-b gene shows a broad range of neutrality indices across mammalian taxa, and explore the biological factors that can explain this observation. We find that observed patterns of differences in the neutrality index, polymorphism, and divergence are not caused by differences in mutational bias. They can, however, be explained by a combination of a small fraction of neutral amino acid sites, weak selection acting on most amino acid mutations, and differences in effective population size among taxa.

Compared to nuclear loci, mitochondrial genes face a number of special circumstances—uniparental inheritance (Gyllensten et al. 1985; Birky 1995), little or no recombination (reviewed in McVean 2001), strong intracellular purifying selection (Stewart et al. 2008), unusual patterns of mutation (Lynch et al. 2006; Haag-Liautard et al. 2008; Montooth and Rand 2008), and potential heteroplasmy (Birky 1995; Haag-Liautard et al. 2008). These factors are all likely to influence their patterns of evolution and variation (Rand and Kann 1996; Lynch and Blanchard 1998; Rand 2001; Neiman and Taylor 2009). Indeed, Weinreich and Rand (2000) showed that mitochondrial loci are consistently different from nuclear loci with respect to the relative values of polymorphism and divergence levels. Specifically, mitochondrial loci tend to have a high neutrality index, NI—the ratio of nonsynonymous to silent polymorphism, divided by the ratio of nonsynonymous to silent divergence—indicating an excess of nonsynonymous polymorphisms relative to divergence. Such an excess is not expected under neutrality, where NI values should be close to 1. More recent analyses, using larger datasets than were available to Weinreich and Rand (2000), cast doubt on whether high NI values are typical for invertebrate mitochondrial loci, but still show a wide range of NI among species of mammals and birds, including some very high values (Rand and Kann 1996; Hasegawa et al. 1998; Nachman 1998; Bazin et al. 2006; Lynch et al. 2006; Berlin et al. 2007; Meiklejohn et al. 2007; Nabholz et al. 2008). For example, in the Berlin et al. (2007) study of the cyt-b gene, values of the neutrality index for mammals range from 1.1 to 11; values for birds are generally even higher.

Here, we develop a new population genetic model, based on that of Loewe et al. (2006) and Haddrill et al. (2010), to ask what factors might explain the broad range of NI values at mitochondrial loci in mammals. Models of selection acting on nonsynonymous sites have examined the effect of different probability distributions of selection coefficients on polymorphism and divergence (reviewed in Eyre-Walker and Keightley 2007). Our approach is close to that of Welch et al. (2008), but uses a simpler method to generate the theoretical predictions. As in these previous models, we model the effect of differences in effective population size among species, and allow for a distribution of selection coefficients across nonsynonymous sites. However, the model developed here also allows us to explicitly model two additional biological factors that potentially have large effects on mitochondrial NI values—(1) a fraction of nonsynonymous sites with no selective effects and (2) mutational bias away from the allele favored by selection. This allows us to produce a more realistic spread of potential NI values than previous models, and hence provides a better fit to the data.

Model

To model neutrality indices at mitochondrial loci, we calculate theoretical predictions of between-species divergence values and polymorphism levels for neutral and selected sites, assuming a probability distribution of γ, the scaled selection coefficient against a deleterious mutation. For a given site under selection, we have γ= 4Nes, where s is the selection coefficient against a deleterious nonsynonymous mutation (Kimura 1983, p. 44), and 2Ne is the mean time to coalescence for a neutral site (Charlesworth and Charlesworth 2010, p. 217). At each nonsynonymous site, we allow for mutations to and from a single favored variant, and among its alternatives, each of which is assumed to be equally deleterious. Over all sites, there is an equilibrium flux of substitutions among the favored and deleterious variants caused by mutation, selection, and genetic drift, which contributes to both polymorphism and divergence levels (Bulmer 1991; McVean and Charlesworth 1999; Tachida 2000; Loewe et al. 2006; Haddrill et al. 2010). To relate these predictions to observed polymorphism and divergence at nonsynonymous and synonymous sites, we assume that synonymous sites are neutral. We also assume that there is a fraction, cn, of nonsynonymous sites that are neutral, whereas the remaining nonsynonymous sites are subject to purifying selection (cn may of course take a value of zero). To model the distribution of γ values, we use either a gamma or a log-normal distribution, following previous practice (Loewe et al. 2006; Welch et al. 2008; Haddrill et al. 2010).

RATES OF SUBSTITUTION FOR SELECTED AND NEUTRAL SITES

To model selected sites, we use a modification of the formulation of Haddrill et al. (2010), which is based on that of Loewe et al. (2006). We assume a sequence of m nucleotide sites, each with two alternative types: m1 sites are fixed for the selectively favored A1 allele, and m2 sites are fixed for the selectively deleterious A2 allele. As in Loewe et al. (2006), we consider a favored A1 allele that can mutate to selectively equivalent, deleterious variants at rate μ, collectively denoted by A2. We divide the mutation rate at the site fixed for the deleterious alternatives into two components: ζμ for the rate of mutation back to the favorable variant, and ξμ for the rate of mutation to any of the selectively equivalent deleterious amino acids. The reason for describing the two types of back mutation rates by two independent parameters, rather than representing them by ζμ and (1 −ζ)μ as in Haddrill et al. (2010), is that the base content for favored amino acids may differ from that of the deleterious alternatives at a given codon. Because mutation rates vary among different bases—with a strong bias toward transitions over transversions and GC>AT versus AT>GC mutations in mitochondrial DNA (e.g., Aquadro and Greenberg 1983)—the mutation rates depend on the particular variant present at a site. Hence, either of the ζ or ξ parameters could be greater or less than one, depending on the base composition of nonsynonymous sites.

At statistical equilibrium, despite continual changes at individual sites, the expected numbers of sites in states A1 and A2 are constant over time. Thus, the expected number of sites fixed for A1 that eventually experience a substitution to A2, m1λ1, is exactly balanced by m2λ2 fixations in the reverse direction, where λ1= (2Nμ) u(γ), λ2= (2Nζμ) u(−γ), and u(γ) is the fixation probability for a mutation with scaled selection coefficient γ. (We use 2N as the population size, rather than N, to make our notation consistent with the standard diploid case.) To calculate the overall expected number of substitutions at selected sites, we must also take into account the exchange among the selectively equivalent deleterious alleles at the A2 sites, of which 2Nm2ξμ arise every generation with neutral fixation probabilities, 1/(2N).

To express the rate of substitution at selected sites in terms of scaled selection coefficients, γ, we use standard population genetic equations, following Loewe et al. (2006) and Haddrill et al. (2010). We assume that Neμ is sufficiently small that most sites are fixed for either A1 or A2. Under our assumptions concerning forward and back mutations for a given set of mutation and selection parameters, the equilibrium proportion of sites fixed for the favorable variant, x*=m1/m, is given by the Li–Bulmer equation (Bulmer 1991; McVean and Charlesworth 1999), where ζ is equivalent to the reciprocal of the mutational bias for favorable to unfavorable changes that is usually used; this gives x*=ζ/{ζ+ exp(−γ)}. To obtain the λ values referred to above, we use the standard equations for fixation probabilities with haploid selection (Fisher 1930; Kimura 1962, 1983, p. 45). For a new copy of a mutation, the approximate probability of fixation for a weakly selected deleterious mutation is u(γ) = 2s(Ne/N){exp(γ) − 1}, and for the reverse mutation it is u(−γ) = 2s(Ne/N){1 − exp(−γ)}.

To remove dependency on the mutation rate, we divide the substitution rate at selected sites by that at neutral sites, which is assumed initially to be equal to μ. Averaging over sites fixed for A1 and A2, and integrating over the distribution of selective effects of new mutations, φ(γ), we obtain the net rate of substitution at selected sites relative to that at neutral sites for a given value of the mutational bias parameter ζ (Haddrill et al. 2010, eq. A6):

image(1)

LEVELS OF POLYMORPHISM AT SELECTED AND NEUTRAL SITES

Each generation, a certain number of fixed sites experience a mutation, contributing to nucleotide site diversity. At equilibrium, these changes are balanced by segregating sites that become fixed, so that the mean level of diversity remains unchanged. We write π1* for the expected nucleotide site diversity for sites at which the ancestral state is A1; π2* is the corresponding value for sites with ancestral state A2. A formula for these is given by Charlesworth and Charlesworth (2010, p. 278). To obtain the approximate equilibrium diversity at selected sites, π*, we average over both classes of sites, weighting π1* by x* and π2* by ζ(1 −x*). We also include a contribution 4Neμξ(1 −x*) from mutations among selectively equivalent deleterious alleles, because 4Neμ is the expected neutral diversity.

Again, we wish to express these quantities relative to the neutral values, and in terms of a distribution of scaled selection coefficients. Using the same mutation model as above for a given γ and averaging over A1 and A2 sites, then dividing by the neutral value 4Neμ and carrying out some algebraic simplifications, gives the relative diversity at selected sites for a given value of ζ as:

image(2)

Integrating over a distribution of selection coefficients, φ(γ), as for the divergence values above, yields:

image(3)

POLYMORPHISM AND DIVERGENCE AT NONSYNONYMOUS AND SYNONYMOUS SITES

We wish to relate these calculations of the theoretical values for selected and neutral sites, such as Ksel/Kneut, to the corresponding observed values for nonsynonymous and synonymous sites, so that KA/KS. As stated above, synonymous sites are assumed to be neutral. As it seems reasonable to suppose that there may also be a small fraction of nonsynonymous sites for which substitutions are neutral, we incorporate this feature into the model as well. In principle, this class of amino acid sites might strongly affect the value of the neutrality index, as they will contribute disproportionately to both polymorphism and divergence, causing NI to tend toward the neutral value of 1. If cn is the fraction of amino acid sites that are neutral, the relative site diversity (averaging over a distribution of selection coefficients) becomes:

image(4)

The equation for the theoretical value of KA/KS is analogous: we integrate inline image over the distribution of scaled selection coefficients. The theoretical value of the neutrality index is given by the ratio of these two quantities, inline image.

The application of equations (1–4) to nonsynonymous and synonymous sites assumes that there is no difference between mutation rates μNS and μS for nonsynonymous and synonymous sites, respectively. This, of course, is incorrect, because of differences in base composition among classes of site: coding positions that experience nonsynonymous mutations in mammals tend to be more GC-rich than other coding positions; in the cyt-b dataset used here, the mean GC content for the mostly nondegenerate first and second coding sites is GC12= 0.426, and the mean of GC3 for degenerate third coding sites is 0.378 (Mann–Whitney U= 7124.5, P= 8.9 × 10−7). The well-known GC to AT mutational bias for mtDNA (e.g., Aquadro and Greenberg 1983) thus leads to a higher mutation rate for nonysynonymous mutations than synonymous mutations, so that our equations will underestimate π*A*S and K*A/K*S.

We therefore correct for unequal mutation rates, using plausible values for the mutation rate parameters derived from data on the base compositions of the different classes of site. We assume that the GC content at fourfold degenerate sites (about 30%; see also Perna and Kocher 1995) reflects the equilibrium value due to mutational bias alone, giving κ= 2.3 for the ratio of the GC>AT to AT>GC mutation rates, because the equilibrium GC content under neutrality is equal to 1/(1 +κ) (Bulmer 1991).

First, we estimate the mutation parameters for nonsynonymous sites. For mammalian mitochondrial genes, only zero- and twofold degenerate sites are relevant, because no deleterious variant is possible at fourfold degenerate sites, and the genetic code for mammalian mitochondrial genes has no threefold degenerate codons. Furthermore, at twofold sites, the only amino acid changing mutations are transversions; because these are very rare in mitochondria (Brown et al. 1982, Aquadro et al. 1984), we can, to a first approximation, ignore them, so that we need only take zerofold sites into account.

At zerofold sites, 41% of nucleotides are GC, and we assume that this largely reflects the composition of the optimal alleles at these sites. Then, at these sites, the forward mutation rate, μNS, from the favored to the deleterious state, is 0.41κu+ 0.59u= 1.53u, where u is the mutation rate to transitions at AT sites. For sites fixed for the deleterious allele, the relevant mutation rates depend on the base composition of the deleterious alleles. Given the assumption that the deleterious alternatives to a favorable allele are selectively equivalent, the relative frequencies of GC versus AT among the deleterious alleles reflect the products of the frequencies of the favorable states from which they arose and the mutation rates away from these, that is, 0.41κu and 0.59u for GC and AT, respectively. This means that the mutation rate back to the favorable variant, ζμNS, is equal to (0.41κ+ 0.59κ)u/(0.41κ+0.59) =κu/(0.41κ+ 0.59) = 2.3u/(0.41 × 2.3 + 0.59) = 1.5u. Because there are no selectively equivalent deleterious transition variants at zerofold sites, ξμ= 0 at these sites.

We compare these rates to the rate at fourfold synonymous sites; because these are 30% GC, they experience a net mutation rate of 0.3(2.3u) + 0.7u= 1.39u. Thus, the ratio of μNS to μSYN is not 1, as implicitly assumed above, but close to 1.53/1.39 = 1.1; we correct the values for the equations for π*A*S and K*A/K*S above by multiplying by this constant (note that there will be no effect on NI, as this constant will occur in both the numerator and denominator). It follows that ζ and ξ are equal to (1.50/1.53) = 0.98 and (0/1.53) = 0, respectively, and we use these values in our calculations.

ADAPTIVE EVOLUTION

Adaptive evolution affecting amino acid sites will elevate KA/KS, thus reducing the NI value. We can incorporate a fraction of sites fixed by positive selection into our model. If the proportion of nonneutral sites experiencing adaptive mutations is ca, and their rate of fixation relative to that at neutral sites is λa, the relative rate of substitution at nonsynonymous sites becomes:

image(5)

Let the mean equilibrium nonsynonymous diversity relative to synonymous diversity (again integrated over all possible selection coefficients) in equation 3 be π*d. The ratio of nonsynonymous to silent diversity is then:

image(6)

where 2caλa represents the upper limit to the contribution of an adaptive substitution to polymorphism during its sojourn in the population (Fisher 1930). The ratio of these two equations gives a modified, more general, expression for NI. It is worth noting that the proportion of amino acid substitutions due to positive selection (α) is related to the quantities just described, by α= (caλa)(KA/KS).

Given the limited number of observations, we cannot estimate ca and λa as well as the other parameters of interest. Instead, we examine a range of plausible values for caλa. Fortunately, the plausible range of this quantity is quite limited, as it cannot be higher than KA/KS, and there is little evidence that adaptive evolution strongly influences NI values in mammalian mitochondria (Bazin et al. 2006; Mulligan et al. 2006; Nabholz et al. 2008).

MODELING THE DISTRIBUTION OF SELECTION COEFFICIENTS

We considered two different distributions of scaled selection coefficients (γ), the gamma and log-normal distributions. Both distributions have been widely used in the literature (e.g., Piganeau and Eyre-Walker 2003; Loewe et al. 2006; Welch et al. 2008; Haddrill et al. 2010), and we therefore have some idea of which parameter estimates are consistent with polymorphism and divergence data. (We nevertheless used a broad range of parameters, to check that our results are robust with regard to the choice of parameter values.)

The gamma distribution is given by:

image(7)

with mean inline image, shape parameter β (equal to the reciprocal of the square of the coefficient of variation), and a scale parameter =β/inline image (Kimura 1979; Welch et al. 2008). The denominator contains the gamma function, inline image

The log-normal distribution is given by:

image(8)

with mean μ on the scale of the natural logarithm of γ, and shape parameter σg= exp(σ), which is the standard deviation on a log scale.

DATA ANALYSIS

We analyzed alignments of the mitochondrial cytochrome b gene (cyt-b), downloaded from the Polymorphix database (http://pbil.univ-lyon1.fr/polymorphix; Bazin et al. 2005). We removed alignments with stop codons, corrected misalignments, and removed sequences that did not cover the entire gene. We used DNAsp (Librado and Rozas 2009) to calculate pairwise estimates of amino acid and synonymous site diversity (πA and πS), which were Jukes–Cantor corrected (Jukes and Cantor 1969). Divergence estimates were obtained using the KaKs_calculator (Zhang et al. 2006), based on the Goldman–Yang method (Goldman and Yang 1994) and applying a standard HKY-85 correction (Hasegawa et al. 1985) to correct for unobserved substitutions. When multiple outgroups were available, we chose the closest as measured by KS. To facilitate comparisons with the theoretical predictions, the observed values of NI were obtained as (πAS)/(KA/KS), rather than using the numbers of synonymous and nonsynymous polymorphisms and fixed differences, as is usually done. However, the significance of NI was assessed using a Fisher's exact test on raw counts of divergent and polymorphic sites. A total of 82 species were analyzed; for one of the 82 taxa, πAS and NI values could not be estimated because of zero πS, and these were excluded from analyses involving these quantities.

Statistical analysis were performed using R (http://www.r-project.org); Tarone's test for heterogeneity was used as implemented in the metafor package (Viechtbauer 2010).

Results and Discussion

FACTORS AFFECTING NI

We used the model developed above to examine the effect of several factors on NI: the choice of distribution used to model γ values, the mean and shape of the distribution, and the magnitude of fixed parameters, for example, the fraction of neutral amino acids (cn) and the mutational bias (ζ). We calculated theoretical values of πAS, KA/KS, and NI separately for two distributions of γ values, the gamma and log-normal; Figure 1 shows the specific parameterizations of these distributions that were used. To examine the effect of each of the remaining parameters, we changed its magnitude while holding the remaining parameters constant. Each of these cases is discussed in turn below.

Figure 1.

This shows the gamma and log-normal distributions used to model the fitness effects of mutations (in terms of γ= 4Nes). The plots show the probability densities of γ for distributions with a mean of 200 and several different values of the shape parameter. The distributions are (A) the gamma distribution [with shape parameters β shown on the plot, corresponding to coefficient of variations (CV =β−0.5) of 3.2, 2.2, 1.4, 1.0, and 0.7, with the smallest shape parameter corresponding to the largest CV]; (B) the log-normal distribution [with median and geometric mean μg= exp(μ) = 200, and shape parameters σg= exp(σ) corresponding to CV values of 0.23, 0.42, 0.79, 14.1, and 2104.8 (CV = (exp[σ2]− 1)0.5; the smallest σg corresponds to the smallest CV)].

First, the effect of mean γ on NI when mean γ is relatively small can be understood qualitatively by considering the expected values of NI for different classes of selected sites: (1) neutral sites, where NI= 1; (2) sites under strong purifying selection, which contribute little to both polymorphisms and substitutions; (3) sites under weak purifying selection, which contribute more to polymorphism than to divergence, and thus yield NI > 1. The effect of the mean of γ on NI values can then be understood intuitively. For example, for a mean γ near 0, most sites are neutral, and NI is close to 1; as the mean of γ increases away from 0, most amino acid mutations contributing to NI are weakly selected, and NI exceeds 1 (Figs. 2 and 3).

Figure 2.

Predictions of population genetic parameters for different means and shape parameters of the gamma distribution of mutational effects on fitness. In all cases, other parameters were fixed at cn= 0.005, ζ= 0.98, and ξ= 0. The y-axes in the panels correspond to predicted values of: (A) πAS (B) KA/KS (C) NI. The horizontal dashed lines in this and all following plots represent the 5% and 95% quantile for observed values of the statistic of interest, calculated from a dataset of 83 cyt-b genes (see text). Although the plots show expectations only for a mean γ of 1000 and below, calculations show that the expectations are near this asymptotic value, at least until mean γ= 10,000.

Figure 3.

Predictions of population genetic parameters when mutational effects are modeled using the log-normal distribution. In all cases, other parameters were fixed at cn= 0.005 and ζ= 0.98 and ξ= 0. The y-axes in the panels correspond to predicted values of: (A) πAS, (B) KA/KS (C) NI. Again, calculations show that the predictions remain at approximately the asymptotic value at least until mean γ= 10,000.

Second, the effect of the shape parameter can be understood by considering the contribution of slightly deleterious mutations. When the fraction of these mutations is large—for example, with small mean γ and a shape parameter reflecting a narrow distribution—NI indices are large, as expected when there is an excess of slightly deleterious polymorphism (see the curves for large β in Fig. 1 and for small σ in Fig. 2). A wide distribution, on the other hand, tends to result in small NI values even for large mean γ, because there is then a long tail of strongly selected amino acid sites, which contribute little to NI when cn > 0 (Figs. 2 and 3; see the argument in the next paragraph).

Third, inclusion of a fraction of neutral amino acid mutations has a strong qualitative effect on NI values (Fig. 4). With cn= 1, both amino acid and synonymous sites are neutral, and NI= 1; for cn= 0 (not shown), NI increases monotonically with mean γ, as in Welch et al. (2008). The more interesting cases are those in which cn takes some intermediate value (0 < cn < KA/KS). In these cases, the distribution of NI values can be strongly modal, with a maximum value at a finite mean γ(Figs. 2–4). The reason is that the contribution of weakly selected versus neutral amino acid sites changes with mean γ. For small mean γ, the contribution from weakly selected sites (for which NI > 1, as described above) increases with mean γ. As the mean increases even further, however, most selected sites become strongly selected, and thus do not contribute to either polymorphism or divergence. NI is then mostly determined by the fraction of neutral nonsynonymous sites, and therefore tends back toward 1. The size of the resulting mode depends on the magnitude of cn (Figs. 2–4). In addition, note that with small cn, NI tends to be greater than 1 for much of the parameter space; as cn increases, NI tends back toward 1, due to the larger contribution from neutral nonsynonymous sites (Fig. 4).

Figure 4.

Effect on the neutrality index of cn, the fraction of nonsynonymous sites that evolve neutrally. The y-axes show the predicted values of NI for cases in which the fitness effects of nonneutral mutations are modeled with (A) gamma and (B) log-normal distributions. The ζ and ξ parameters are fixed at 0.98 and 0, respectively, and the shape parameters shown are β= 2 (plot A; top group of lines) and β= 0.1 (plot A; bottom group of lines; lines for different cn values indistinguishable) and σg= 1.5 (plot B; top group of lines) and σg= 10 (plot B; bottom group of lines).

Finally, we analyzed the effects of two factors that do not appear to contribute to the variation in NI. The value of the mutational bias parameter ζ, perhaps surprisingly, has very little influence on the magnitude of NI; different values of ζ lead to very similar values of NI, as can be seen by the closeness of the lines in Figure 5. The inclusion of sites experiencing adaptive evolution in the model depressed NI values, as would be expected intuitively (results not shown).

Figure 5.

Effect of varying ζ, the mutational bias parameter. The y-axes show the predicted values of NI for cases in which the fitness effects of nonneutral mutations are modeled with (A) gamma and (B) log-normal distributions. The cn parameter is fixed at 0.01, and the shape parameters are, as before, β= 2.0 (plot A; top group of lines) or β= 0.1 (plot A; bottom group of lines), and σg= 1.5 (plot B; top group of lines) or σg= 10 (plot B; bottom group of lines). The lines correspond to ζ values of 0.75, 1.0, 1.5, and 2.0, but are not labeled because of their similarity.

DISTRIBUTION OF NI VALUES IN MAMMALS

We confirmed that mammalian neutrality indices cover a broad range of values by reanalyzing a dataset of cyt-b genes from 82 mammalian species (Nabholz et al. 2008). Most NI values—, 83%, or 68 of 82—are larger than 1, although only 16 of these are significantly larger than 1 at the 5% level (or only three after a Bonferroni correction). However, the range of values is broad (NI ranges from 0 – 11.6, excluding one value that could not be estimated due to πS= 0), and, overall, NI is significantly larger than 1 (median NI= 2.16; Wilcox test for NI greater than 1, W= 3137, P= 1.83 × 10−12). Further, the distribution of NI values is broad, ranging from 0–20.1 (Fig. 6), showing significant heterogeneity (Tarone's test for heterogeneity, Tarone 1985, χ2= 257.36, df = 79, P < .0001, excluding two taxa for which both KA and KS equal 0).

Figure 6.

The y-axes show the observed values from a dataset of mitochondrial cyt-b genes from 83 species for (A) πAS, (B) KA/KS (C) NI.

Taking these data at face value, then, the trend in mammals appears to be toward high NI values, as has been observed previously (e.g., Berlin et al. 2007; Meiklejohn et al. 2007; Neiman and Taylor 2009). However, two biases can affect NI values, which can result in inaccurate estimates. First, a large synonymous divergence between the ingroup and outgroup taxa can cause KS to be underestimated, and hence NI to be undererestimated, which is reflected in the negative correlation that can occur between divergence and NI under neutrality (Wares et al. 2006). We tried to minimize this effect by using the closest outgroup taxa available and by correcting divergence values for multiple hits; in our dataset, the correlation between NI and KS is weakly positive, rather than negative (rS= 0.222, P= 0.045), and NI values remain significantly above 1 for both high and low KS (values below median KS, W= 744, P= 4.79 × 10−7; values above, W= 836, P= 4.11 × 10−10).

Second, as pointed out by Stoletzki and Eyre-Walker (2011), estimates of NI are statistically biased because the expectation of a ratio of sample statistics is not equal to the ratio of their expected values, leading to an overestimation of the true NI value. When we calculate values of their alternative, unbiased, statistic (DoS), however, the resulting estimates are consistent with our NI values: both NI and DoS indicate that the predominant force affecting cyt-b is purifying selection (DoS ranges from −1.85 to −0.035, excluding two values for which it could not be calculated). The DoS statistic does not lend itself as readily as our version of NI to calculations of its theoretical value; thus, we have confined ourselves to NI.

The 16 NI values that are individually significantly larger than 1 at the 5% level range from 1.13 to 20.1. Although some of these NI values may be poorly estimated due to small sample sizes, some are estimated from large samples (e.g., estimates of 4.5 and 6.1 are obtained from taxa with sample sizes of 73 and 22, respectively) or with large numbers of polymorphisms (e.g., an estimate of NI= 4.4 from a taxon with 116 synonymous polymorphisms). This high range of values is interesting, as the calculations above suggest that NI values much above three are possible only under a limited range of parameter values that are also consistent with the low observed πAS and KA/KS values. (Note that taxa with NI significantly >1 do not have πAS and KA/KS values notably different from those for other taxa, πAS= 0.020–0.278 and KA/KS= 0.004–0.045 for high NI taxa vs. πAS= 0–0.248 and KA/KS= 0.005–0.409 for others). In particular, only distributions with a low mean gamma and a narrow distribution of effects, that is, those in which many mutations have weak effects, are consistent with observed values of πAS and KA/KS and NI taken together (see Figs. 2 and 3 with low mean γ and large β or small σ). This remains true provided that the fraction of neutral amino acid sites is reasonably low (less than half the KA/KS value, Fig. 4).

If we allow for a moderate increase in mean γ (e.g., as a consequence of a two- to threefold increase in Ne), distributions that generate a high value of NI for a given value of mean γ can also yield NI values near one, which represent the majority of estimates in the data (Figs. 2 and 3; note that with smaller cn, smaller changes in Ne will also suffice). This is particularly the case if γ values are log-normally distributed (Fig. 3). Thus, the data are consistent with the combination of a narrow distribution selective effects (i.e., in which most amino acid mutations are under weak purifying selection), and modest differences in effective population size between taxa. This is also consistent with the proposal of Berlin et al. (2007) that Hill–Robertson effects (Hill and Robertson 1966) caused by association with the W chromosome reduce the Ne for the mtDNA of birds, as, compared to mammals, birds have higher NI values for cyt-b.

Given the results of our modeling, we conclude that the broad range of NI values in mammals probably arises from the combination of (1) a large fraction of nearly neutral amino acid mutations, (2) a small fraction of neutral amino acid sites, and (3) differences in Ne between taxa. Consistent with this idea, species with high NI values show low πS (with the data divided into thirds, πS-low= 0.100, πS-mid= 0.144, πS-high= 0.048). Although such a difference might also be due to a recent elevation in mutation rate in the low NI taxa, so that πS has recently been elevated while KS remains mostly unaffected, this possibility is ad hoc and seems unlikely. In fact, the idea that elevated NI values are associated with reduced Ne is not a new one, as mitochondrial loci were historically observed to have higher NI than nuclear loci, and are also expected to have lower Ne due to their uniparental inheritance, haploidy, and potential to suffer from strong Hill–Robertson effects due to their lack of genetic recombination (Rand and Kann 1996; Lynch and Blanchard 1998; Rand 2001; Neiman and Taylor 2009). As we have shown, this explanation is, however, only viable if there is some fraction of nonsynonymous mutations that are neutral (cn > 0), because otherwise NI increases monotonically with mean γ (Welch et al. 2008), so that lower Ne due to Hill–Robertson effects would lead to lower rather than higher NI values.

Although we have developed a model that can explain the range of NI values observed for mammalian cyt-b on the basis of plausible differences among taxa in Ne, it is possible that there are other contributing factors that are not included in this model. Two possibilities—heteroplasmy and selection acting on silent sites—are unlikely to greatly affect neutrality indices. Although heteroplasmy, in which individuals harbor more than one mitochondrial type, does occur in mammals, it is a transient phenomenon occurring on a much shorter time scale than genetic drift (Birky 1995; Haag-Liautard et al. 2008), and so can have little effect on genetic diversity. Selection acting on synonymous sites also does not appear to affect mitochondrial sequences: instead, the base content of synonymous sites reflects strand-specific biases (Min and Hickey 2007), and in cases where genes have moved from one strand of the mitochondrial genome to the other, synonymous sites change to reflect the base composition bias of the new strand (Asakawa et al. 1991). This strongly suggests that it is the neutral mutational bias, and not selection, that determines base composition at synonymous sites (Perna and Kocher 1995). In any case, population genetic calculations suggest that the effect of weak selection on neutrality indices is small (Haddrill et al. 2010).

The model used here assumes a simple demographic model of a single panmictic population of constant size. In reality, mitochondrial diversity may often be affected by population growth, because mitochondrial diversity data from mammals often show a pattern consistent with recent population expansion, as evidenced by their generally negative Tajima's D values (Wares 2010). Alternatively, such values may be produced by Hill–Robertson effects in a nonrecombining genome (Kaiser and Charlesworth 2009; Seger et al. 2010). The effect of population size growth on NI is unclear: high current population sizes should generally lead to lower NI once equilibrium at the new larger population size is attained, but ongoing population growth may also cause transient elevations of πA relative to πS and hence higher NI values (Travis et al. 2007). In this dataset, however, there is no relationship between Tajima's D at synonymous sites and NI values (rs= 0.026, P= 0.817), suggesting that population expansion is not the cause of high NI values.

The data analyzed here are also potentially affected by population substructuring: most of the studies here have sampled across the geographic range of the species, and so diversity estimates may be affected by limited migration and genetic differentiation across subpopulations (Nabholz et al. 2008). The following argument suggests, however, that population structure is unlikely to explain the overall elevation of NI values above 1. First, although population structure can cause fixation probabilities to differ from those used above to calculate expected NI values, this effect is largely mediated through dominance (unless the population structure deviates strongly from a conservative migration model: Nagylaki 1982; Wakeley 2003). (Because mitochondria are effectively haploid, dominance is not relevant here.) Second, the NI values that are difficult to explain are the high values, and the overall effect of population structure will usually be to reduce NI. The reason is that population subdivision will primarily act to increase elevate πA/πS compared to a panmictic population, as neutral diversity will be elevated by population substructuring, unless there are strong effects of local extinction and colonization for which there is relatively little evidence (Pannell and Charlesworth 2000); sites subject to purifying selection will be relatively unaffected. Because fixation probabilities are little affected by population structure, it should have a minimal effect on KA/KS, resulting in an overall reduction in NI.

Conclusions

Using an equilibrium model of nucleotide substitution and polymorphism, we have investigated forces affecting neutrality indices at mitochondrial loci. We find that the range of NI values in mammals is easiest to explain by a combination of a distribution of mutational effects that includes a large proportion of weakly selected mutations, as has been previously been argued by others (reviewed in Neiman and Taylor 2009), variation in effective population size, and a small fraction of neutral amino acid sites.


Associate Editor: J. Wares

ACKNOWLEDGMENTS

We thank K. Brown and J. Welch for helpful discussions, and two anonymous reviewers and A. Eyre-Walker for their comments on the manuscript. AJB was supported by a grant to BC from the Biotechnology and Biological Sciences Research Council of the UK.