Copyright � 2009 by the Genetics Society of AmericaDOI: 10.1534/genetics.109.105692

Measuring the Rates of Spontaneous Mutation From Deepand Large-Scale Polymorphism Data

Philipp W. Messer1

Department of Biology, Stanford University, Stanford, California 94305

Manuscript received May 31, 2009Accepted for publication June 9, 2009

ABSTRACT

The rates and patterns of spontaneous mutation are fundamental parameters of molecular evolution.Current methodology either tries to measure such rates and patterns directly in mutation-accumulationexperiments or tries to infer them indirectly from levels of divergence or polymorphism. Whileexperimental approaches are constrained by the low rate at which new mutations occur, indirectapproaches suffer from their underlying assumption that mutations are effectively neutral. Here I presenta maximum-likelihood approach to estimate mutation rates from large-scale polymorphism data. It isdemonstrated that the method is not sensitive to demography and the distribution of selectioncoefficients among mutations when applied to mutations at sufficiently low population frequencies. Withthe many large-scale sequencing projects currently underway, for instance, the 1000 genomes project inhumans, plenty of the required low-frequency polymorphism data will shortly become available. Mymethod will allow for an accurate and unbiased inference of mutation rates and patterns from such datasets at high spatial resolution. I discuss how the assessment of several long-standing problems ofevolutionary biology would benefit from the availability of accurate mutation rate estimates.

NUCLEOTIDE mutations are the ultimate source ofgenetic variation within populations and between

species. Mutations initially occur in individuals, yet somemight subsequently become fixed in the population.Such substitution events underlie the evolution of species.Precise knowledge of the rates and patterns of sponta-neous nucleotide mutation is hence of essential impor-tance for our understanding of the evolutionary process.

The characteristics of mutations can be analyzedby mutation-accumulation experiments (Luria andDelbrck 1943; Denver et al. 2004; Haag-Liautard

et al. 2008; Lynch et al. 2008). These approaches areconfined, however, to experimentally feasible organ-isms. Their accuracy is also limited by the generally lowrate at which new mutations occur in individuals. Muta-tion patterns might furthermore be peculiar in specificanalyzed strains. An accurate estimation of mutation ratesand patterns on local genomic scales by mutation-accumulation experiments is clearly beyond the scopeof present-day experimental capabilities.

For practical purposes, one therefore often usesindirect approaches to investigate mutation character-istics. Indirect approaches are based on predictions frompopulation genetics theory that quantitatively link themutational processes to the expected levels of divergencebetween species or polymorphism within a population.

They typically rely on the assumption that mutations areeffectively neutral.

For example, population genetics theory predicts thatthe amount of polymorphism in a population is relatedto the quantity u ¼ 4Nem, where m is the rate ofspontaneous mutation in an individual genome andNe is the effective population size. A variety of differentestimators for u from polymorphism data exist (Ewens

2004), but all either depend on the neutrality assump-tion or require explicit knowledge about the distribu-tion of selection coefficients among new mutations.

Polymorphism-based approaches are also particularlysensitive to demography, especially when they utilizepolymorphism data over the full range of populationfrequencies. A population bottleneck, for instance, canremove a large amount of polymorphism from thepopulation. Mutation rates estimated from the amountof polymorphism under the assumption of constantpopulation size might then substantially underestimatethe true rates (Tajima 1989b).

Population genetics theory also links mutations andsubstitutions. Here it is predicted that substitution ratesequal mutation rates if mutations are effectively neutral(Kimura 1968). The rates and patterns of substitutionbetween species can thus provide a proxy for rates andpatterns of mutation in individuals under the aboveassumption.

Such divergence-based approaches underlie most of ourpresent estimates of mutational parameters (Nachman

1Address for correspondence: Department of Biology, Stanford University,371 Serra Mall, Stanford, CA 94305. E-mail: messer@stanford.edu

Genetics 182: 1219–1232 (August 2009)

and Crowell 2000; Kumar and Subramanian 2002;Ellegren et al. 2003). Primarily this might be due tothe greater availability of divergence compared to poly-morphism data. Divergence-based analyses should alsobe less affected by demography than polymorphism-based approaches, yet they rely more crucially on theassumption of selective neutrality.

The widespread acceptance of divergence-based ap-proaches relates to Kimura’s influential ‘‘neutral theoryof molecular evolution,’’ which surmises that mostsubstitutions and observed polymorphisms are indeedeffectively neutral (Kimura 1968). On a genomewidescale, the effects of selection on the population dynam-ics of new mutations can hence safely be neglected, evenmore so when restricting analyses to presumably un-constrained regions of genomes like pseudogenes,inactivated transposable elements, or fourfold degen-erate codons.

In recent years the neutral theory has been stronglychallenged. There is accumulating evidence that inmany species selection is far more prevalent thanpreviously thought (Fay et al. 2002; Andolfatto 2005;Bustamante et al. 2005; Eyre-Walker 2006; Begun

et al. 2007; Nielsen et al. 2007; Macpherson et al. 2007;Cai et al. 2009). In addition, biased gene conversion(BGC), which with regard to allele frequency dynamicsoperates identically to selection, seems to be acting inmany higher organisms (Nagylaki 1983; Galtier andDuret 2007). In the light of such evidence it remainsquestionable to what extent indirect approaches thatmeasure levels of divergence or polymorphism at ‘‘pre-sumably neutrally’’ evolving sequence regions may stillprovide accurate estimates for the true rates and patternsof mutation.

In principle, many of the biases that result whenmutations are not effectively neutral should vanishwhen utilizing polymorphism data at very low popula-tion frequencies. This is because the population dy-namics of low-frequency alleles are predominantlygoverned by stochastic, rather than selective forces. Inthis regime, all mutations should behave similarly, irre-spective of their particular selection coefficients. More-over, mutations at low population frequencies shouldalso be less affected by past demographic events becauseon average they are younger than mutations at higherfrequencies (Kimura and Ohta 1973).

A more intuitive example of why low-frequencymutations should become less sensitive to both selec-tion and demography is to consider the extreme limit ofmutations that are present in only one individual of thepopulation. These mutations are likely to have justoccurred in the parental germline. They will neither beinfluenced by the species demography nor be influ-enced by selection—except for dominant lethals. Onecan therefore expect such mutations to reflect the‘‘true’’ rates and characteristics of the underlying muta-tional processes.

Single-nucleotide polymorphism (SNP) data at suffi-ciently low population frequency should hence allow forthe inference of rates and patterns of spontaneousnucleotide mutation in a way that is less affected by thedistribution of selection coefficients among new muta-tions and the particular demographic history of thespecies. Such an approach has not been feasible so fardue to the lack of genomewide SNP data at the requiredlow population frequencies.

This restriction will shortly be overcome. Severallarge-scale sequencing projects are presently beingconducted, for example, the 1000 genomes project inhumans (Kaiser 2008). These experiments will providelarge amounts of genomewide SNP data at sufficientlyhigh population resolution, finally making the regimeof low-frequency variation accessible for quantitativeinvestigation. To utilize such data for an unbiasedinference of mutational parameters, one requires esti-mators that isolate particular frequency classes from thefrequency spectrum of SNPs.

Here I develop a maximum-likelihood (ML) methodfor measuring u from the observed numbers of SNPs atparticular population frequencies. When applied to low-frequency SNPs, it allows for an unbiased inference ofmutation rates and patterns at high regional resolutionand accuracy. The method does not require priorknowledge of the distribution of selection coefficientsamong new mutations or the demographic history of thespecies. I demonstrate that ML estimates always convergeto the true rates as long as investigated populationfrequencies are sufficiently low. Analytical formulas forthe deviations between ML estimates and the true rates inthe presence of selection and demography are alsoprovided, and it is discussed how these deviations, inturn, can be used to infer selection and demography.

The method is expected to yield accurate and robustmutation rate estimates from the anticipated SNP datasets. For the 1000 genomes project in humans I estimatethat the expected spatial resolution of the methodshould allow for a regional inference of mutation rateson genomic length scales ,100 kbp.

The availability of regionally resolved rates and patternsof spontaneous mutation would encourage the assess-ment of many important problems in evolutionary biology(Duret 2009). Examples include the elucidation of therelative contributions of drift and selection to evolution,the investigation of extent and characteristics of BGC, andthe characterization of inherent biases of mutation pro-cesses. The application of my ML approach and itspotential advantages over substitution-based approachesfor such analyses is discussed at the end of this article.

BACKGROUND

The aim of this study is to establish a ML methodologyfor inferring the rates of spontaneous mutation fromthe numbers of low-frequency SNPs in polymorphism

1220 P. W. Messer

data sets. To compute likelihoods of observed countsgiven particular mutational parameters one requires aprobabilistic model of the expected numbers of suchcounts. The starting point for this probabilistic modelis the expected frequency distribution of mutations inthe source population from which the SNP data havebeen obtained. Fortunately, an analytic formula forthis distribution already exists. It is discussed in thefollowing.

Let us consider a panmictic population of N diploidindividuals. Mutations are characterized by their selec-tion coefficients s, and codominance is assumed. Indi-viduals heterozygous for a mutation have fitness 1 1 s,homozygotes have fitness 1 1 2s, and individuals withoutthe mutation have fitness 1. Mutations are modeledaccording to an infinite-sites model. Mutations withselection coefficient s arise in individuals by a Poissonprocess with rate mg, where g ¼ 2Ns determines thestrength of selection associated with a mutation. Differentmutations evolve independently of each other. Segregat-ing sites in the population can be classified according tothe g’s of their mutant alleles. For each class g, we definewith gg(x) the expected average number of segregatingsites in the population at which the mutant allele ispresent at population frequency x, the so-called sitefrequency spectrum (SFS). Under mutation-selectionequilibrium, Wright (1938) has shown that

ggðxÞ ¼ 2mg

12e22gð12xÞ

ð12e22gÞxð12xÞ : ð1Þ

The SFS can also be deduced from Kimura’s seminaldiffusion approximation for the stochastic dynamics ofallele frequencies in a population under the influenceof random genetic drift and selection (Kimura 1964;Sawyer and Hartl 1992; Ewens 2004). As this frame-work will prove instructive for my further analysis, it isshortly outlined here. Let f(p, x, t) be the conditionalprobability density that a mutation from class g is atfrequency x in the population at time t, given that itsinitial frequency is p at time t ¼ 0. The stochasticdynamics of f(p, x, t) per generation in the diffusionapproximation are then determined by

@f

@t¼ 1

4N

@2

@x2 ½xð12xÞf�22g@

@x½xð12xÞf�

� �: ð2Þ

The first term on the right-hand side describes thestochastic influence of random genetic drift on f(p, x,t), and the second term specifies the average determin-istic rate of change in x due to selection. In the limit oflow frequencies x, Equation 2 is dominated by the driftterm and the relative contribution of selection to thevariance in allele frequency between generations be-comes negligible. Consequently, one can also expect thedistribution (1) to converge to its neutral asymptoticsgg/0(x) for small x. Indeed, Taylor series approxima-tions yield

ggðxÞ ��!g/0 2mg

x[ g0ðxÞ and ggðxÞ ��!x/0

g0ðxÞ: ð3Þ

Examples for the rate of convergence can be seen inFigure 1, where distributions gg(x) are shown for severalexemplary values of g.

From the SFS for class g one can calculate theexpected overall number of segregating sites in thepopulation from that class,

mgðmgÞ ¼X

x2XN

ggðxÞ: ð4Þ

Here the sum is taken over all possible frequenciesin a diploid population of N individuals, XN ¼f1=ð2N Þ; . . . ; ð2N 21Þ=ð2N Þg. The normalized distri-bution of mutant frequencies is then

rgðxÞ ¼ ggðxÞ=mgðmgÞ: ð5Þ

Note that rg(x) does not depend on mg because bothnumerator and denominator are proportional to themutation rate.

It needs to be pointed out that the diffusion approx-imation, and thus the SFS derived from it, are valid onlyin a restricted regime of population parameters. Thisregime is specified by the conditions N ?1 and jg j>N .The latter condition implies that lethal and semilethalmutations cannot be treated in terms of the diffusionapproximation. Such mutations therefore need to beexcluded from further analysis.

Figure 1.—Expected number of mutant alleles present atfrequency x in a population. Distributions gg(x) are shown forseveral different selection classes, always using mg ¼ 1. Thesolid line is the neutral asymptotics, g0(x) ¼ 2/x, which inthe double-logarithmic plot appear as a straight line withslope �1. Compared with neutral mutations, deleterious mu-tations (g , 0) are systematically suppressed from reachinghigher frequencies in the population, and beneficial muta-tions (g . 0) are enriched at high frequencies. In the low-frequency limit all distributions converge to the neutralSFS, although convergence occurs substantially faster for ben-eficial than for deleterious mutations.

Measuring Mutation Rates 1221

RESULTS

In principle, the rate of spontaneous mutation can becalculated from the SFS (1) by measuring gg(x) at givenfrequencies x, provided that one knows the selectioncoefficients of mutations and that the population can beassumed to be in mutation–selection equilibrium.

Both prerequisites will often not be fulfilled. SNPfrequency data are typically obtained from genomicregions for which one has no prior knowledge aboutthe distribution of selection coefficients among newmutations. And the SFS can substantially deviate frommutation–selection equilibrium for nonstationary de-mographic histories. In addition, one also has to accountfor possible sampling biases resulting from the fact thatSNP frequency data will be estimated from only a sampleof genotyped individuals from the population.

In this section I first describe a ML approach to infermutation rates mg for SNPs from a given selection class g

that assumes mutation–selection equilibrium, yet ac-counts for sampling biases due to a finite number ofsequenced strains. I then show how this approach can beapplied to SNP data from mutations with an unknowndistribution of selection coefficients by restricting theanalysis to very low-frequency SNPs. Quantitative ex-pressions for the expected errors are also derived. Ifinally discuss how my method is affected by a breach ofmutation–selection equilibrium due to demographicforces such as recent population expansions and bottle-necks. It is demonstrated that the influence of de-mography on my estimates is effectively reduced to onlyvery recent population size changes when focusing onlow-frequency SNPs.

ML estimation of mg for a given selection co-efficient: Let us assume a sequence region was geno-typed in n individual haploid genomes. The truefrequency of a mutation from class g in the populationthat is observed in k of n genotyped sequences will notbe exactly x¼ k/n. Instead, x will be specified in terms ofa probability distribution. One can calculate this distri-bution via Bayes’ theorem,

Prðx j k; nÞ

¼ Prðk j x; nÞPrðxÞPx92XN

Prðk j x9; nÞPrðx9Þ ¼Bxðk j nÞrgðxÞP

x92XNBx9ðk j nÞrgðx9Þ :

ð6Þ

Here I used rg(x) from Equation 5 as a prior. Bx(k j n)denote binomial distributions to incorporate the effectsof sampling. In Figure 2, Pr(x j k, n) is shown for neutralmutations and exemplary values of k. Depending on thevalue of g, the true population frequency of a mutationcan be substantially overestimated by simply using x ¼k/n as a proxy for x.

The denominator of (6) defines the marginal prob-ability to observe a mutation from class g in k of ngenotyped sequences,

P kg [ Prðk j n; rgÞ ¼

Xx2XN

rgðxÞBxðk j nÞ: ð7Þ

Let Gkg be the measured overall number of mutations

from class g that are observed in k of the n genotypedsequences. The probability to observe Gk

g , given thatmg(mg) sites are segregating in the population, is thenagain a binomial distribution. This probability defines alikelihood function for the underlying mutation rate mg,

LkgðmgÞ ¼ Pr½Gk

g j mg� ¼ BP kg½Gk

g j mgðmgÞ�: ð8Þ

By maximizing Lkg(mg) over mg the ML estimate for the

data can be derived. One can also measure values Gkg for

a set of sample frequencies, k 2 K, and calculate LK forthe entire set,

LKg ðmgÞ ¼Yk2K

LkgðmgÞ: ð9Þ

Here it is assumed that likelihoods Lkg for different k are

independent of each other, which should be a reason-able approximation as long as n>N .

ML estimation of m for arbitrary distributions ofselection coefficients: From Equations 8 and 9 one canderive ML estimates of mg from measuring counts ofmutations from class g in a sample of n genotypedindividuals. This approach is of limited practicalitybecause SNPs in a given sequence region will comprisemutations with several different selection coefficients,the distribution of which we are unlikely to have priorknowledge of. When measuring the number of muta-tions present in k of n genotyped sequences, an overallcount for mutations from all different classes of selec-tion coefficients will be obtained,

Gk ¼X

g

Gkg : ð10Þ

The rate of spontaneous mutation in the investigated re-gioncan bedefinedbysummingoverall individual rates mg,

Figure 2.—Probability density Pr(x j k, n) that a neutralmutation has population frequency x in a population of sizeN¼ 104 if it is observed in k of n¼ 1000 genotyped sequences.

1222 P. W. Messer

m ¼X

g

mg: ð11Þ

The true likelihood function for m is again a binomialdistribution. Formally it is given by

LkðmÞ ¼ BP k Gk j mðmÞ� �

: ð12ÞHere m(m) is the (unknown) expected number ofsegregating sites and P k is the (unknown) probabilityto find a mutant allele in k of n sequences at a seg-regating site.

In the following I show that even without knowledge ofthe particular distributions of selection coefficients, andthus the precise values of m(m) and P k, one can still inferaccurate ML estimates of m by restricting the analysis tomutations at low population frequency (k>n) andapproximating Lk(m) by the neutral-likelihood function

Lk0ðmÞ ¼ BP k

0½Gk j m0ðmÞ� with P k

0 ¼X

x2XN

r0ðxÞBxðk j nÞ: ð13Þ

Mathematically it is not immediately obvious that thisneutral approximation always works. After all, bothparameters m0(m) and P k

0 of the neutral-likelihoodfunction can substantially differ from their true valuesm(m) and Pk if selection coefficients are not zero. Forexample, if many mutations are deleterious, then therewill be fewer segregating sites compared to the neutralexpectation. One will therefore overestimate the ex-pected overall number of SNPs in the population byusing m0(m) as a proxy. The SFS at those sites, on theother hand, will be skewed toward smaller frequenciescompared to the neutral expectation. Hence one willunderestimate the probability to observe a mutation atlow frequency at a given segregating site. In the nextparagraph I show analytically that both deviationscompensate for each other in the limit x / 0.

Let us assume that selection coefficients among newmutations are distributed according to an (unspecified)distribution v ¼ {vg} in terms of the individual ratios vg ¼mg/m.Tostartmyderivation, Ifirstpointout thatanaccuratecalculation of Lk and L0

k relies on large-enough numbersGk. The expectation value of Gk can be calculated by

hGki ¼ mðmÞP k ¼ u

4N

Xg

mgðvgÞP kg ; ð14Þ

with u ¼ 4Nm. The corresponding expectation of Gk

under the assumption of neutrality yields

hGk0 i ¼ m0ðmÞP k

0

¼ m0ðmÞX

x2XN

r0ðxÞn

k

� �xkð12xÞn2k

��!N ?14N m

ð1

0

n

k

� �xk21ð12xÞn2kdx

¼ u

k: ð15Þ

In the third line I exchanged the summation over allfrequencies from the set XN by an integral over the

interval [0, 1], which is feasible if N ?1. Note that hGk0 i is

independent of the number of genotyped strains.The low-frequency asymptotics of the likelihood

functions (12) and (13) are technically obtained byevaluating L1(m) and L1

0(m) in the limit n / ‘, and Ihence require that hG1i?1 and u?1. In this regime, thecentral limit theorem states that the binomial distribu-tion L1

0ðmÞ ¼ BP 10½G1 jm0ðmÞ� converges to a normal

distribution with mean m0P 10 ¼ u and variance

m0 P 10 ð1 2 P 1

0 Þ � u. Accordingly, the true-likelihoodfunction L1(m) will converge to a normal distributionwith mean and variance hG1i. A normal distribution isunambiguously defined by its mean and variance. Toprove that the true-likelihood function always convergesto the neutral-likelihood function in the low-frequencylimit, it thus suffices to show that

limn/‘hG1i ¼ u: ð16Þ

To calculate the limit let us first consider products ofthe form

mgðvgÞP 1g ¼

Xx2XN

mgðvgÞrgðxÞnxð12xÞn21

���!N ?14N vg

ð1

0

12e22gð12xÞ

12e22g nð12xÞn22dx

¼ 4N vgð12e22gÞ21 n

n212

ð1

0e22gð12xÞnð12xÞn22dx

� �:

ð17ÞThe last integral can be expressed in terms of incom-plete gamma functions, G½a; x� ¼

Ð ‘

x ta21e2tdt,ð1

0e22gð12xÞnð12xÞn22dx

¼ nð2gÞ12n Gðn21; 0Þ2Gðn21; 2gÞ½ �

��!n/‘e22g: ð18Þ

This result applies for arbitrary vg; therefore

limn/‘hG1i ¼ u

4N

Xg

4N vg ¼ u: ð19Þ

From the central limit theorem it then immediatelyfollows that

limn/‘

L1ðmÞ ¼ limn/‘

L10ðmÞ: ð20Þ

I have shown above that the true-likelihood functionconverges to the neutral-likelihood function in the limitx / 0 for arbitrary distributions of selection coeffi-cients. One can hence expect ML estimates derived bythe neutral-likelihood function from low-frequencySNPs in a given genomic region to approximate thetrue rates for that region. If observed values Gk aresufficiently large, the neutral-likelihood function con-verges to a Gaussian distribution

Measuring Mutation Rates 1223

Lk0ðuÞ ¼

1

ðu=kÞffiffiffiffiffiffi2pp exp 2

Gk2ðu=kÞ� �2

2ðu=kÞ2

!: ð21Þ

For a given k, my ML estimator for u will hence be of thesimple form

uðkÞ ¼ kGk for 0 , k , n: ð22Þ

When both alleles at a polymorphic site are counted, i.e.,the folded spectrum Gk is measured, then the estimatoruðkÞ is consistent with the expectation value hGki ¼u½1=k 1 1=ðn2kÞ� derived in Tajima (1989a). Note thatfor unfolded spectra, uðkÞ does not depend on theoverall number n of genotyped strains, yet the expectederror of uðkÞ resulting from nonneutral mutations will.The magnitude of such errors is calculated below. Forneutral mutations the estimator is correct for all k andalso consistent with Watterson’s commonly used estima-tor uw (Watterson 1975), which is based on the overallnumber S ¼

Pn21k¼1 Gk of segregating sites observed in a

sample of n genotyped sequences,

uw ¼SP

n21k¼1 1=k

¼P

n21k¼1 uðkÞ=kP

n21k¼1 1=k

: ð23Þ

Sensitivity to selection: What will be the error of theestimator uðkÞ if mutations are not neutral? My MLapproach provides a straightforward way to calculate theexpected error for an assumed distribution v of selectioncoefficients: Analogously to Equation 21 one can alsoapproximate the true-likelihood function (12) by aGaussian distribution. Its mean and variance are givenby hGki. From Equations 14 and 15 it then follows that

uðkÞu¼ k

2m

Xg

ð1

0ggðx; mgÞBxðk j nÞdx: ð24Þ

In Figure 3 values of the expected relative errorsuðkÞ=u are shown for different strengths of selection anddifferent values of k in an assay of 1000 genotypedsequences. For simplicity it is assumed that all mutationshave the same selection coefficient g. Hence, thedistribution v of selection coefficients has only onenonzero value vg¼ 1. Figure 3 confirms the expectationthat the relative error of uðkÞ increases for more negativeselection coefficients and higher sample frequencies k.But it will still be sufficiently small for practicable samplefrequencies as long as selection is not too strong. Forexample, when estimating uðkÞ at k ¼ 5 with n ¼ 1000,the true rate will be underestimated by ,10% for g ¼�10 and still only �40% for g ¼ �50. Deviations due topositive selection are very small and limited by an upperbound that does not depend on the actual strength ofselection.

In Figure 4 full-likelihood curves Lk0ðmÞ are shown for

several mutation scenarios. I thereby first calculated, for

a given mutation scenario v, the average number hGkiof mutants one expects to observe in k of n samplesaccording to Equation 14. From rounded values Gk ¼roundhGki neutral-likelihood curves Lk

0ðmÞ were calcu-lated as defined by Equation 13. The maxima of thelikelihood curves approach the correct mutation rate ask becomes smaller. Errors uðkÞ=u accurately coincidewith the values predicted by Equation 24.

Equation 24 also allows one to calculate the expectederror of the estimator uðkÞ if selection coefficientsamong new mutations are specified in terms of theirdistribution. However, the shape of this distribution ismuch debated and hypotheses vary widely (Keightley

1994; Fay et al. 2001; Nielsen and Yang 2003; Piganeau

and Eyre-Walker 2003; Yampolsky et al. 2005; Boyko

et al. 2008). Clearly one will also expect distinctdistributions for different species and different classesof mutational events. Nonsynonymous mutations, forexample, are likely to have different distributions ofselective effects than synonymous mutations (Akashi

and Schaeffer 1997). And mutations in noncodingregions will again differ from both of the above.

For calculating error bounds of uðkÞ due to non-neutral mutations one does not require full knowledgeof the distribution of selection coefficients. It suffices tohave upper limits for one or more of its quantiles. Wehave already seen in Figure 3 that positive selection willnot significantly influence the error. Hence, if we knowthat maximally a fraction dg of new mutations is moredeleterious than a particular g , 0, then the limitfor the expected error can be calculated by uðkÞ=u ¼ðk=2Þ

Ð 10 ggðx; mg ¼ 12dÞBxðk jnÞdx according to Equa-

tion 24. This means one would very conservativelyassume that a fraction 1 � d of mutations has selectioncoefficient g, and the remaining mutations would be sodeleterious that they are never observed as SNPs.Information on additional quantiles can be incorpo-rated analogously.

Figure 3.—Expected relative errors uðkÞ=u according toEquation 24 for nonneutral mutations as a function of gfor three different k ¼ 2, 5, and 10 in a sample of n ¼ 1000genotyped sequences.

1224 P. W. Messer

Sensitivity to demography: Demographic events cancause substantial deviations of the SFS from its equilib-rium shape. A recent population expansion, for exam-ple, will lead to a SFS that is skewed toward lowerfrequencies because new mutations that emerged afterthe expansion have not yet had enough time to reachhigher population frequencies (Slatkin and Hudson

1991). Population bottlenecks can substantially reducethe overall number of polymorphic sites in a populationand lead to a more uniform SFS. Bottlenecks andexpansions are common demographic patterns in sev-eral species, including Drosophila melanogaster (Li andStephan 2006; Thornton and Andolfatto 2006) andhuman (Harpending et al. 1998). It is thereforeessential to investigate how the SFS, and consequentlymy ML estimates, are affected by such demographicevents.

For neutral polymorphism, the expected shape of theSFS in demographic histories with population sizechanges can be calculated analytically following theapproach outlined in Williamson et al. (2005). The key

idea is to segment the demographic history into asequence of time intervals where population size isconstant within each interval, but changes instanta-neously between intervals. The demographic history isthen specified by the sequence N1, N2, . . . , Nn ofpopulation sizes in the successive intervals and thenumbers of generations t1, t2, . . . , tn each intervallasted.

The transition probability density fi(p, x, ti) that anallele, initially present at population frequency p at theend of stage i� 1, has frequency x at the end of stage i isgiven by the transient solution of Equation 2 for g¼ 0. Ithas been calculated by Kimura as

fðp; x; tiÞ

¼X‘

i¼1

4ð2i 1 1Þpð12pÞiði 1 1Þ Ti21ð122pÞTi21ð122xÞe2iði11Þti=ð4Ni Þ:

ð25ÞHere Ti�1(x) are Gegenbauer polynomials, which canbe defined in terms of hypergeometric functions,Ti�1(x) ¼ (i/2)(i 1 1)F[i 1 2, 1 � i, 2, (1 � x)/2].See, e.g., Crow and Kimura (1970) for a discussion ofEquation 25 and its derivation.

New mutations arise at rate 2mNi during stage i andhave initial population frequency 1/(2Ni). We canexpress gi(x), the SFS at the end of stage i, as a functionof gi�1(x) at the end of stage i � 1 plus the contributionof new mutations that entered the population duringstage i,

g iðxÞ ¼X

p2XNi21

g i21ðpÞfðp; x; tiÞ2Ni

1 m

ðti

0f

1

2Ni; x; t

� �dt:

ð26Þ

This allows for an iterative calculation of the present-dayg(x) given an initial g0(x), usually chosen as the SFS at onepoint in the past when equilibrium was assumed to hold.Note that the farther back in time g0(x) lies, the lessinfluence its particular shape will have on g(x). Especiallythe most relevant low-frequency part of g(x) will begoverned predominantly by recent mutations and hencewill be less affected by ancient demographic events.

Kimura also succeeded in deriving analytic solutionsfor the diffusion Equation 2 in the presence of selection(Crow and Kimura 1970), but expressions for thetransition probabilities become very complex. However,for small frequencies x the diffusion equation is alwaysdominated by the drift term, and so will be the transitionprobabilities f(p, x, t) when both p and x are small. Theinfluence of selection on g(x) should therefore becomenegligible for small x irrespective of the particulardemographic scenario.

If the precise demographic history of a population isknown, one can obtain the expected present-day SFSg(x) by iterative application of Equation 26. In practice,though, estimates of the demographic history of apopulation are often unknown or at least surrounded

Figure 4.—Neutral-likelihood curves Lk0ðmÞ for several mu-

tation scenarios. The mutation rate is always m ¼ 0.005. Thedistribution of selection coefficients for a particular scenariois specified by the parameters vg. The expected counts of mu-tations to be observed in k of n ¼ 1000 samples, hGki, wereestimated from Equation 14 for each mutation scenario. Like-lihoods Lk

0ðmÞ were calculated according to Equation 13. Thesize of the source population was N ¼ 104.

Measuring Mutation Rates 1225

by considerable uncertainty. How accurate will it be insuch cases to apply the simple estimator uðkÞ fromEquation 22 to infer the present-day value uc ¼ 4Ncm

with the contemporary population size Nc? For a givendemographic scenario one can easily calculate theexpected relative error by

uðkÞuc¼ k

2m

ð1

0g ðxÞBxðk j nÞdx: ð27Þ

Note the structural analogy to Equation 24, where therelative error of uðkÞ under constant population size butin the presence of selection was calculated.

I investigated the magnitude of the expected error forthree prominent demographic scenarios to show thatuðkÞ provides accurate estimates for uc when evaluatedat small k. The first model (Figure 5A) is a scenariosuggested for the African-American human subpopula-tion that features an instantaneous population growth(Boyko et al. 2008). The second and third models aretwo scenarios proposed for the European D. melanogastersubpopulation. The model of Li and Stephan (2006)supposes an ancient population expansion followed bya severe population bottleneck (Figure 5B). The compar-atively simpler model of Thornton and Andolfatto

(2006) supposes only a population bottleneck (Figure5C). All three demographic scenarios can be expectedto yield present-day SFS that substantially deviate fromthe equilibrium of Equation 3.

I numerically estimated the expected present-day SFSg(x) for the three demographic scenarios by performingextensive forward simulations. For practical application,the simulation approach turns out to be more efficientcompared to a semianalytical approach based on thecalculation of Equation 26 because the infinite sums inthe transition probabilities (25) converge only veryslowly if initial frequencies p are small.

For my simulations I assumed that the SFS was inequilibrium (3) at an ancient point in time (dotted linesin Figure 5, A–C). Expected numbers of segregatingneutral sites at that time, m0, and their normalizedfrequency distribution were calculated according toEquations 3–5 for a chosen value of m. Then m0 siteswere drawn randomly from the frequency distributionand their trajectories were simulated by binomialsampling under a Wright–Fisher model for the partic-ular demographic scenario.

New mutations arising in stage i were modeled by aPoisson process with rate 2mNi. Their frequency dynam-ics were also simulated by binomial sampling startingfrom the respective initial frequencies p¼ 1/(2Ni). Theexpected present-day SFS g(x) for each demographicscenario was then obtained by combining the present-day frequencies of all simulated segregating sites. Ichose values m¼ 10.0 for scenario A, m¼ 1.0 for scenarioB, and m ¼ 0.01 for scenario C, which resulted insufficiently large numbers of segregating sites to ap-proximate g(x) in each scenario with high accuracy.

The simulation algorithm was implemented in C11.Runs were performed on up to 250 CPUs of the Bio-X2

cluster at Stanford University. All software is availablefrom the corresponding author upon request.

In Figure 5D error ratios uðkÞ=uc according toEquation 27 using the numerically estimated g(x) areshown for the three demographic models. As expected,uðkÞ converges to the correct uc in all three demographicscenarios if k is chosen sufficiently small. For scenarioA, the proposed demographic model for African-American humans, the relative error will be ,5% whenestimated at k ¼ 5. Errors for the two demographicmodels of the European D. melanogaster subpopulationare larger, but still converge to the correct present-dayestimates for small enough k.

Figure 5.—Analysis of the expected errors uðkÞ=uc for threeexemplary demographic scenarios. The time arrows in thethree scenarios (A–C) go from the past to the present (tipsof the arrows). Intervals of constant population size are spec-ified by their respective population sizes Ni and durations ti.Population size changes instantaneously between intervals.Dotted lines specify ancient points in time at which an equi-librium SFS was assumed. The sample size in D was n ¼ 1000.

1226 P. W. Messer

DISCUSSION

Low-frequency polymorphism contains valuable in-formation on the characteristics of mutational pro-cesses. At low population frequencies, the dynamics ofderived alleles closely resemble those of effectivelyneutrally evolving mutations. Low-frequency alleleswill also be comparatively young. Methods that infermutational parameters from low-frequency variationshould thus be less affected by selective and demo-graphic effects compared to divergence-based meth-ods or those based on the full frequency spectrum ofpolymorphism.

Deep and large-scale SNP data sets comprising suffi-cient numbers of sequenced individuals to allow for acomprehensive analysis of low-frequency SNPs will shortlybecome available. Hence there is clearly a need for analysismethods that can focus on genetic variation at particularpopulation frequency classes for such inference.

I presented a ML method for the estimation ofmutation rates from polymorphism data that can beapplied to every frequency class separately. My approachworks by comparing the measured counts Gk of muta-tions that are present in a particular number k of ngenotyped sequences to their expectation for a givenunderlying mutation parameter u in terms of the simpleneutral estimator uðkÞ ¼ kGk. It can be applied specifi-cally to low-frequency SNPs, and above I showed that theneutral approximation is valid for this regime. This waymy ML approach does not require prior knowledge ofthe distribution of selection coefficients among newmutations and, in addition, becomes less sensitive topast demographic events.

Error sources and their evaluation: The expectederrors of the estimator u in a practical analysis can bedivided into four categories: (i) stochastic errors due tosampling, (ii) errors resulting from inaccuracy of theSNP data set, (iii) SNP polarization errors, and (iv)systematic errors due to violation of my assumptions.They are discussed in order.

i. Stochastic sampling errors are fully incorporated inmy likelihood analysis. The magnitude of sucherrors can be derived by calculating confidenceintervals around ML estimates from the likelihoodfunction (13) or its Gaussian approximation (21).

ii. Data set inaccuracies will primarily result from se-quencing errors or misalignment. They can lead towrongly identified or missed SNPs and incorrect es-timation of SNP frequencies in the sample (Hellmann

et al. 2008; Lynch 2008; Liu et al. 2009). Theresulting errors in my ML estimates will be de-termined by the probability of such errors in thedata set. One can substantially reduce their magni-tude by disregarding singletons (k¼ 1) or setting aneven higher threshold for the minimum k used inthe analysis. For example, assuming a sequencingerror rate of 10�5, a genome size of 3 3 109, and 1000

sequenced genomes, the expected number of in factnonpolymorphic sites that are erroneously identi-fied as being polymorphic with k ¼ 4 would be onthe order of only one.

iii. It has been assumed so far that all SNPs in the dataset are perfectly polarized; i.e., for every polymor-phic site we have exact knowledge of which is thederived allele and which is the ancestral allele.Although such information can in principle beobtained from comparison with an out-group spe-cies, it might be prone to error. However, given thatmy analysis intends to focus on variation at very lowpopulation frequency, it is presumably much saferto simply assume that the low-frequency allele isalways the derived allele and not to refer to an out-group species for such classification.

The expected number of wrongly classified allelesby this approximation can be easily estimated fromthe SFS (1). Let us consider a SNP with minor allelefrequency x and assume that the derived allele isneutral. Then the probability that the derived alleleis actually the one at the larger frequency isg0(1 � x)/[g0(x) 1 g0(1 � x)] ¼ x. And thus thiserror will be small for low-frequency SNPs. Fordeleterious mutations it will be even yet smaller.For beneficial mutations, on the other hand, theerror probability can ultimately become as large as0.5. If a substantial number of beneficial mutationsare expected in the data set, SNP polarization by anout-group species might indeed be advisable.

iv. Systematic errors can arise in my analysis if one ormore of its underlying assumptions are violated.One basic assumption is the applicability of aninfinite-sites model. It might be violated in largeSNP data sets if sites with more than two alleles areobserved. Having decided on a threshold minimumk, alleles that occur in less than the minimum ksequences can simply be masked. In the rare casethat more than two alleles are present at a poly-morphic site above the threshold one can eitherdisregard these sites and estimate the resulting errorfrom the fraction of such sites in the data or treat allindividual low-frequency alleles at one site as in-dependently derived alleles at different sites.

More critical are systematic biases due to selectionor demography. With Equations 24 and 27 I pro-vided analytic expressions for the expected errors ofuðkÞ when the full distribution of selection coeffi-cients among new mutations, respectively the par-ticular demographic history of the species, is known.I quantitatively investigated the magnitude of theseerrors for a wide range of selection coefficients(Figure 3), as well as several prominent demo-graphic scenarios (Figure 5), showing that uðkÞbecomes insensitive to both selection and demog-raphy when estimated at small enough k.

Measuring Mutation Rates 1227

My ML method provides a simple test to check therobustness of the estimator uðkÞ directly from the data.The key observation for this test is that uðkÞ ¼ u shouldbe constant for all k if the underlying assumptions ofneutrality and unvarying population size are sufficientlymet. Both nonneutral mutations and past demographicevents will lead to characteristic biases in uðkÞ thatdepend on k in a systematic manner: In the presence ofmany deleterious mutations, for example, one expectsuðkÞ to decrease with increasing k because the SFS fordeleterious mutations is skewed toward smaller frequen-cies compared to the neutral spectrum. Prevalentpositive selection, on the other hand, should lead to asystematic increase of uðkÞ at larger k. Similar argumentshold for violations of the assumption of constantpopulation size as discussed earlier.

When combining the effects of demography and se-lection, complex interactions can arise. Yet it is highlyunlikely that selective and demographic effects com-pensate for each other in a way that makes the present-day SFS appear unaffected by both. Therefore, if nostrong systematic changes of uðkÞ are observed for the datawhen varying k, assumptions are most likely appropriate.

Interpretation of uðkÞ for complex demographichistories: From uðkÞ one obtains estimates of the rates ofspontaneous mutation only in terms of the populationparameter u¼ 4Nm, as is typical for methods that utilizepolymorphism data for such inference. Absolute valuesof m thus cannot be obtained, unless one knows theprecise value of N. This raises the question which N theestimator refers to, especially in the presence of com-plex demographic histories.

In population genetic analyses this problem is usuallytackled by the introduction of an effective populationsize, Ne, specifying the actual rate of change of allelefrequencies in the population due to random geneticdrift. Effective population sizes are influenced by avariety of factors, including population substructure,selection, and demography (Charlesworth 2009).Often Ne is much lower than the current number ofindividuals in a species (Frankham 2007).

If only effects of demography are taken into accountand we assume that all sites in a genome evolve in-dependently of each other, then Ne for neutral variationcan be expressed in terms of the demographic historyN(t) of the species. Here N(t) is the actual number ofindividuals in the species at time t, measured backward intime from t ¼ 0 at present. The effective population sizefor a neutral allele that emerged t generations ago will begiven by the harmonic mean of N(t) over its time ofexistence (Charlesworth 2009),

Ne ¼ t

ðt

01=N ðtÞdt

� �21

: ð28Þ

Note that the harmonic mean of N(t) over an interval isdominated by its smallest values in that interval. The

average age of a derived allele, however, is itself afunction of population frequency x and effective pop-ulation size (Kimura and Ohta 1973), determined by

tðxÞ ¼ 4NelogðxÞ121=x

: ð29Þ

The effective population size corresponding to a de-rived allele present at population frequency x willtherefore be a function Ne(x). It can be obtained bysimultaneously solving the combined system (28) and(29) for the given demographic history N(t).

When estimating uðkÞ at different k we are comparingsegregating sites at different population frequencies x.The effective population size corresponding to a small kwill hence not be affected by demographic events thatoccurred more than t(k/n) generations ago. SNPs atpopulation frequency x ¼ 0.5%, for example, will haveNe � Nc if the population size did not change sub-stantially from its contemporary size during the last0.1 3 Nc generations.

The relation between k and the corresponding Ne alsoexplains why decreasing ratios uðkÞ=uc are observed forthe three demographic scenarios of Figure 5; they allfeature smaller population sizes in the past comparedwith current sizes Nc. Low-frequency SNPs have not‘‘felt’’ these smaller population sizes, whereas thepopulation dynamics of SNPs contributing to uðkÞ atlarger k might be entirely dominated by the smallerpopulation sizes in the past. In fact, for the simple two-stage scenario A, the estimator uðkÞ converges preciselyto 4Nam for large k (data not shown).

Background selection and selective sweeps: Besidesdemography, also other evolutionary forces can de-crease the effective population size. Adaptive substitu-tion events, for instance, can lower Ne for SNPs in theirgenomic vicinity as a consequence of linkage disequi-librium (Kaplan et al. 1989). A similar reduction is ex-pected to result from backgroundselection (Charlesworth

et al. 1993). In contrast to demography, which shouldaffect Ne genomewide, background selection and selec-tive sweeps can cause local variation of Ne along agenome.

Again, it holds that the lower the frequency of a SNP,the lower is the probability that it has been affected bysuch events. I illustrate this by a rough calculation forthe expected effects of genetic hitchhiking in Drosoph-ila. Macpherson et al. (2007) estimated that a neutralpolymorphism destined for fixation will, on average,experience two selective sweeps in its genomic vicinity.If we assume a constant population size N, then theaverage time to fixation of a neutral polymorphism is 4Ngenerations (Ewens 2004). One can thus roughlyestimate the rate at which SNPs are affected by sweepsto be 1/(2N) per generation. A SNP at populationfrequency x has then been affected by a sweep withprobability t(x)/(2N). For a frequency x ¼ 0.5% thisyields a probability of only �5%.

1228 P. W. Messer

Inferring selection and demography: The present-day SFS g(x) is a function of the mutation parameter u,the distribution of selection coefficients among newmutations, and the demographic history of the species.I have shown that my estimator uðkÞ allows one to inferaccurate estimates of uc ¼ 4Ncm, which become in-sensitive to selection and demography when estimatedat very low population frequencies. This is becausedeviations between the true SFS and its asymptotic form(3) vanish for small k. At higher population frequencies,such deviations become more and more profound. Theparticular shape of g(x) at larger x should in turn pro-vide information on selection and demography. Variousstudies have used this approach to estimate the distri-bution of selection coefficients among new mutations,the demographic history of species, or both simulta-neously by analyzing observed SFS from populationgenetic data sets (Williamson et al. 2005; Thornton

and Andolfatto 2006; Eyre-Walker et al. 2006; Li andStephan 2006; Keightley and Eyre-Walker 2007;Boyko et al. 2008; Gonzalez et al. 2009).

These approaches suffer from the general problemthat selection and demography can never be unambig-uously inferred from the shape of the SFS alone (Myers

et al. 2008). There are always different distributions ofselection coefficients, or different demographic scenar-ios, that give rise to the same SFS. Moreover, bothselection and demography can lead to similar devia-tions, making it difficult to disentangle their individualcontributions. In practice, inference of selection anddemography from the SFS is therefore usually restrictedto fitting simple parameterized models to the data in aML framework.

Such ML inference of demography or selection isstraightforward to incorporate into my method if onecan parameterize the expected SFS g in terms of thevariables of the particular model to be estimated. Forexample, when assuming constant population size but aparticular distribution v of selection coefficients amongnew mutations that one wants to infer, then g can becalculated by g ðvÞ ¼

Pg

vggg, using the gg defined inEquation 1. The SFS can thus be expressed as a functionof v. From g the expected number of segregating sites,m ¼

Px g ðxÞ, and the normalized distribution r ¼ g/m

can be calculated. Analogously to Equation 12, where gwas parameterized by m and a likelihood function for m

was obtained, the likelihood of a particular distributionv is

LkðvÞ ¼ BP k ½Gk j m� with P k ¼X

x

rðxÞBxðk j nÞ:

ð30Þ

For cases where one wants to infer the parameters of aparticular demographic model and can assume thatmutations are selectively neutral, g can be expressed as afunction of the variables of the demographic model

either analytically according to Equation 26 or numer-ically by simulations. Simulations will clearly be theapproach of choice for analyses where neither constantpopulation size nor neutral mutations can be assumed.

The crucial advantage of my approach is again thecapability to calculate likelihoods for different frequencyclasses separately. This can provide substantial improve-ments to previous approaches, as it allows one to focus onthe particularly informative low-frequency part of theSFS. Consider, for example, the two different scenariosB and C for the demographic history of the EuropeanD. melanogaster subpopulation shown in Figure 5. Bothscenarios cause similar reduction in overall heterozygos-ity in the European population compared to the Africanpopulation because their bottleneck strengths tb/Nb arecomparable. Yet the two models can be clearly distin-guished by the large differences of uðkÞ at small k betweenthem (see Figure 5D).

Focusing on low-frequency SNPs might also beparticularly helpful for disentangling demography andselection. This problem has often been approached bydividing SNPs into two classes, the first comprisingpresumably neutrally evolving SNPs, and the secondcomprising SNPs of which the distribution of selectioncoefficients is to be estimated. The rationale is thatdemography can be inferred from the SFS of the neutralclass, which is then used as a proxy when fittingdistributions of selection coefficients to the SFS of thelatter class (Williamson et al. 2005; Eyre-Walker andKeightley 2007; Boyko et al. 2008). The approachhinges of course on the availability of a set of reliablyneutral SNPs. Often synonymous SNPs are used for thispurpose. But it is presently not clear to what degreesynonymous mutations are indeed selectively neutral(Hershberg and Petrov 2008). At higher frequencies,also small selection coefficients can substantially affectthe SFS, potentially causing misleading demographicestimates.

In my analysis this problem can be addressed bysimply investigating the functional dependence of theestimator uðkÞ on k for the different classes of SNPs tocheck the robustness of assumptions for each class. Thisis illustrated in Figure 6, where theoretically expectedcurves uðkÞ are shown for three classes of sites. The threeclasses could depict, for instance, nonsynonymous(Figure 6, squares), synonymous (Figure 6, triangles),and noncoding SNPs (Figure 6, circles).

From the observed curves we would conclude thatassumptions of neutrality and constant population sizeare robust for noncoding and synonymous mutations atthe investigated frequencies, as indicated by the fact thatuðkÞ does not change substantially as a function of k forboth classes (note that the systematic biases resultingfrom the slightly deleterious selection coefficients fornoncoding and synonymous SNPs are very weak). Thisobservation implies in particular that demography isunlikely to be a major issue for SNPs at these low

Measuring Mutation Rates 1229

population frequencies. And SNPs in all classes shouldhave been subject to the same demographic history ofthe species. In fact, nonneutral SNPs, irrespective ofwhether they are deleterious or beneficial, should beaffected by even fewer demographic events than neutralSNPs because they are on average younger (Maruyama

and Kimura 1974).For nonsynonymous SNPs, on the other hand, we see

a systematic decline of uðkÞ with increasing k, indicatingthat they are under selective constraint. One can thensimply fit Equation 25 to the observed uðkÞ to infer thebest-fitting distribution of selection coefficients amongnonsynonymous mutations for the data.

Spatial resolution: The practical applicability of theestimator uðkÞ relies on sufficiently large counts Gk toreduce finite-sample inaccuracies. We can estimate theexpectation value hGk

0 i for a given genomic regionaccording to Equation 15. The mutation rate m isthereby specified as the rate for the entire investigatedregion. The requirement of a large-enough Gk conse-quently poses a limit on the minimum length of theinvestigated sequence region. Next I calculate roughlythe expected spatial resolution of my method for the1000 genomes project.

A common estimate for the per site mutation rate inhumans is 2.5 3 10�8 per generation (Nachman andCrowell 2000). Obtaining an accurate estimate of theeffective population size is more intricate. Widely usedvalues Ne� 104—�105 times smaller than the actual sizeof the human population at present—presumably re-flect the effects of strong population bottlenecks inancient history. These small estimates might be validwhen averaging over SNPs in all frequency classes, butthey will underestimate the effective population sizeassociated with low-frequency SNPs, unless a bottleneckoccurred so recently that low-frequency SNPs have stillbeen affected by it. According to Equation 29 theaverage age of a derived allele at population frequencyx¼ 0.5% is on the order of 0.1 3 Ne. This corresponds toonly 103 generations for the above estimate of Ne ¼ 104.It is unlikely that humans have experienced a severeenough bottleneck within the last 103 generations thatwould justify the small estimate of Ne ¼ 104 during thisinterval. Let us therefore assume that Ne for SNPs atpopulation frequency x¼ 0.5% is at least on the order of105.

With the above estimates one obtains hGk0 i �

1022 3 L=k, where L is the length of the investigatedsequence region. As was already argued earlier, athreshold k¼ 5 should suffice to eliminate severe biasesdue to sequencing errors from my analysis. For a 100-kbp-long genomic region we would expect to observehG5

0 i � 200 SNPs to be present in 5 of the genotypedsequences and still hG50

0 i � 20 SNPs to be present in 50sequences. This should clearly allow for an accurateestimation of u and its robustness for windows of thegiven size. One would even expect to yield good

estimates of u for windows of size 10 kbp from data ofthe 1000 genomes project, but then the robustnessestimation in terms of measuring uðkÞ for larger k willbecome less accurate. Note also that species with largereffective population size, for instance Drosophila, willgenerally permit even higher spatial resolution.

Application in an evolutionary context: The avail-ability of regionally resolved rates of spontaneousmutation would make a multitude of important prob-lems in contemporary evolutionary genomics accessiblefor quantitative investigation (Baer et al. 2007; Duret

2009). For example, it is not clear at present whether theobserved regional variations in substitution rates alonggenomes mainly reflect regional variations in mutationrates or differing degrees of selection, BGC, and otherforces that influence the probabilities of fixation of newalleles (Eyre-Walker and Hurst 2001; Duret andArndt 2008).

Regional mutation rates could be compared withregional values of various other genomic quantities, e.g.,recombination rate, GC content, nucleosome position-ing, etc. In a partial-correlation analysis determinantfactors for regional variations in mutation rate couldpossibly be elucidated.

The rates of spontaneous mutation are expected todepend on biochemical factors like accessibility of agenomic region to mutagenic influences, error prone-ness during DNA replication, and rate and accuracy ofdamage repair (Baer et al. 2007). As a result, the basicmutation process could in fact turn out to be ratheruniversal and its local rate could be primarily deter-mined by a few basic regional features. A low GCcontent, for example, may make the two DNA strandsmore prone to separate, which could increase themutation rate in GC-poor regions (Frederico et al.1993).

Figure 6.—Example of uðkÞ estimated for three differentclasses of sites. All classes have u¼ 1, but selection coefficientsdiffer between classes. For simplicity, all mutations within aclass are modeled to have the same selection coefficient. uwas then calculated by uðkÞ ¼ ðk=2Þ

Ð 10 ggðx; mg ¼ 1ÞBxðk jnÞdx.

The sample size was n ¼ 1000.

1230 P. W. Messer

It is straightforward within my method to resolvemutation rates into the rates of all 12 possible transitionsbetween nucleotides. A regional analysis between theindividual mutation rates and the corresponding sub-stitution rates should prove informative in many aspects,for instance to identify possible mutational biases,estimate the magnitude of BGC, or test hypothesesabout selection for a particular GC content. My methodcan also be easily extended to mutational processesother than single-nucleotide mutations. Potential ex-amples include DNA insertions and deletions, seg-mental duplications, and insertions of transposableelements.

A combined analysis of divergence and polymor-phism at different population frequency classes shouldprovide insight into the interplay between distinct evo-lutionary forces. Low-frequency polymorphism closelyreflects the rates and patterns of spontaneous muta-tions, while polymorphism at intermediate populationfrequencies is shaped, in addition, by selective con-straints. Substitutions finally comprise least constrainedand also adaptive mutations. Knowing the relative pro-portions of deleterious, neutral, and adaptive mutationsis fundamental for our understanding of the evolution-ary process, yet still much is to be learned about theprecise shape of the distribution of selection coef-ficients among new mutations (Eyre-Walker andKeightley 2007).

The ratio between neutral and adaptive mutations isoften estimated by comparing levels of polymorphismand divergence in McDonald and Kreitman-type analy-sis (McDonald and Kreitman 1991). The underlyingrationale of these tests is that polymorphism observed atintermediate population frequencies should mainlyconstitute neutral variation. Low-frequency SNPs areoften intentionally discarded from such analyses todiminish possible biases due to deleterious mutations(Charlesworth and Eyre-Walker 2008).

My estimator uðkÞ should perfectly complementMcDonald and Kreitman tests by shedding light onthe other side of the spectrum, the amount andcharacteristics of deleterious mutations. This class ofmutations is naturally hidden from divergence-basedestimates. So far, it has been accessible only in mutation-accumulation experiments, along with all the naturallimitations of such analyses. Our present knowledgeabout deleterious mutations is hence rather limited.With assays of at least 1000 genotyped sequences, asanticipated for the upcoming large-scale polymorphismdata sets, estimation of uðkÞ at k ¼ 5 should clearly bewithin reach. I demonstrated that this will allow for areasonably accurate estimation of u that captures .90%of deleterious mutations with g¼�10 and still�60% ofdeleterious mutations with g ¼ �50. One can thereforeexpect to obtain, for the first time, estimates of u frompolymorphism data sets that also comprise a substantialfraction of strongly deleterious mutations.

This work was carried out in the Petrov lab at Stanford University.I thank Dmitri Petrov for helpful discussions throughout the projectand carefully proofreading the manuscript. Two anonymous reviewersprovided constructive comments. This work was supported by a long-term postdoctoral fellowship from the Human Frontier ScienceProgram Organization. Computational resources were provided bythe Stanford Bio-X2 compute cluster, supported by National ScienceFoundation award CNS-0619926.

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