Assessing population structure: F ST and related measures

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Molecular Ecology Resources (2011) 11, 5 18 doi: /j x INVITED TECHNICAL REVIEW Assessing population structure: F ST and related measures PATRICK G. MEIRMANS* and PHILIP W. HEDRICK
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Molecular Ecology Resources (2011) 11, 5 18 doi: /j x INVITED TECHNICAL REVIEW Assessing population structure: F ST and related measures PATRICK G. MEIRMANS* and PHILIP W. HEDRICK *Institute for Biodiversity and Ecosystem Dynamics (IBED), University of Amsterdam, PO Box 94248, 1090GE, Amsterdam, The Netherlands, School of Life Sciences, Arizona State University, Tempe, AZ 85287, USA Abstract Although F ST is widely used as a measure of population structure, it has been criticized recently because of its dependency on within-population diversity. This dependency can lead to difficulties in interpretation and in the comparison of estimates among species or among loci and has led to the development of two replacement statistics, F ST and D. F ST is the normal F ST standardized by the maximum value it can obtain, given the observed within-population diversity. D uses a multiplicative partitioning of diversity, based on the effective number of alleles rather than on the expected heterozygosity. In this study, we review the relationships between the three classes of statistics (F ST, F ST and D), their estimation and their properties. We illustrate the relationships between the statistics using a data set of estimates from 84 species taken from the last 4 years of Molecular Ecology. As with F ST, unbiased estimators are available for the two new statistics D and F ST. Here, we develop a new unbiased F ST estimator based on G ST, which we call G ST. However, F ST can be calculated using any F ST estimator for which the maximum value can be obtained. As all three statistics have their advantages and their drawbacks, we recommend continued use of F ST in combination with either F ST or D.Inmostcases,F ST would be the best choice among the latter two as it is most suited for inferences of the influence of demographic processes such as genetic drift and migration on genetic population structure. Keywords: D, fixation, F-statistics, G ST, heterozygosity, population differentiation Received 30 May 2010; revision received 17 August 2010; accepted 2 September 2010 The trouble with F ST Quantifying population structure using F ST Determining the genetic structure of natural populations forms an important part of population genetics and has many applications in evolutionary biology, conservation, forensics, and plant and animal breeding. The method most frequently used to assess population structure is the calculation of F ST, a summary statistic first introduced by Sewall Wright (1943a, 1965). Wright originally developed his F-statistics as inbreeding coefficients, defined as a correlation between uniting gametes. This was long before the advent of allozymes and other molecular genetic markers, and Wright therefore assumed loci to be biallelic. Like most geneticists of his time, he focused on morphological characters with simple Mendelian inheritance. In fact, his landmark paper on isolation by distance and Correspondence: Patrick Meirmans, Fax: ; its effects on the distribution of genetic variation (Wright 1943a) was directly followed by another paper in which he illustrated his findings with a data set on the distribution of white and blue flowers in Linanthus parryae in the Mojave desert (Wright 1943b). Later, when allozymes were introduced as a convenient marker to assess the genetic diversity of a population, Wright s F ST was adapted for use with multiallelic loci, redefined as a ratio of genetic variances (Cockerham 1973). This led to the development of several statistical frameworks to estimate F ST statistics from small samples from a limited number of populations (e.g. Weir & Cockerham 1984; Nei 1987). Nei s (1987) G ST is a direct expansion of Wright s work and is based on a comparison of the expected heterozygosity (gene diversity) within and among populations. The method of Weir & Cockerham (1984) uses an ANOVA approach to estimate within- and among-population variance components, which are then used to estimate their F ST analogue h. These F ST analogues became widely used for analysing allozyme data, primarily because of their 6 INVITED TECHNICAL REVIEW ability to describe the genetic population structure in a single summary statistic and the direct link between F ST and the rate of gene flow (which we will refer to as the migration rate below). With the discovery of new, more variable, genetic markers, new F ST analogues were developed to take the special properties of these markers into account. Excoffier et al. (1992) used the ANOVA approach of Weir & Cockerham (1984), but performed this on a matrix of squared Euclidean distances between DNA haplotypes, from which the F ST analogue u ST is calculated. Slatkin (1995) developed R ST, which is especially suited for markers with a stepwise mutation model such as some microsatellites. In addition to these new methods, the older G ST and h statistics are still widely used for the analysis of highly variable microsatellite markers. Nowadays, a large number of different marker types are available for population genetic studies, with a large range of allelic diversities, from SNPs that are essentially biallelic to microsatellites that can have over 50 alleles at a single locus (e.g. Peijnenburg et al. 2006). Despite their different diversities, all these markers are analysed with what is basically still the same F ST statistic that was originally developed for biallelic data. Only recently have biologists started to become aware of the limitations of F-statistics for analysing data from highly variable loci (Charlesworth 1998; Hedrick 1999, 2005; Balloux et al. 2000; Balloux & Lugon-Moulin 2002; Long & Kittles 2003; Gregorius et al. 2007; Jost 2008; Gregorius 2010). Dependency on H S When defined as a ratio of genetic variances (Cockerham 1973), F ST and its analogues work by relating the amount of genetic variation among populations to the total genetic variation over all populations. For biallelic markers, this makes sure that F ST is bounded between zero and one, with zero representing no differentiation and one representing fixation of different alleles within populations. For multiallelic markers, however, the maximum possible value is not necessarily equal to one, but is instead determined by the amount of within-population diversity (Charlesworth 1998; Hedrick 1999). The reason for this can be best understood by looking at G ST, which is defined as (Nei 1987) F ST Fig. 1 The maximum possible value of F ST as a function of the expected heterozygosity within-population H S (solid line). The closed circles represent values from 84 species published in Molecular Ecology over the last 4 years (expanded from Siegismund and Heller, 2009). ð G ST ¼ H T H S Þ ; H T where H T is the total gene diversity, and H S is the withinpopulation gene diversity (equal to the expected heterozygosity for diploids). Because the true population value of H T is necessarily bigger than or equal than that of H S and the maximum possible value for H T is one, it follows that the maximum value of G ST equals 1)H S (Charlesworth 1998; Hedrick 1999; Jost 2008). For highly variable loci, this can lead to a very small possible range of G ST values. To illustrate this relationship, Fig. 1 gives the joint values of F ST and H S found in the past 4 years in Molecular Ecology (expanded from Heller & Siegismund 2009; see also Table S1, Supporting information). Notice that the observed range of F ST is always less than H S and that the range of F ST becomes very small when H S is large. For example when H S = 0.9, a value that is commonly encountered for microsatellite markers, the maximum possible value of F ST is 0.1. Such a value of F ST is generally interpreted as representing a rather weak population structure. However, here it represents the case with maximum differentiation among the populations, meaning that the populations do not share any alleles at all. It is important to realize that this is not a statistical issue, deriving from the sampling of individuals from populations, but that the problem also occurs when the actual population allele frequencies are used (Jost 2008). Although the dependency on the amount of withinpopulation variation has mostly been discussed for G ST (Hedrick 2005; Jost 2008; Ryman & Leimar 2008), it is also present for other F ST estimators such as h and u ST (Balloux et al. 2000; Meirmans 2006). For these statistics, the calculation of the maximum possible value is less straightforward than for G ST and requires calculating the maximum possible among-population variance component, given the within-population variance in the sample (Meirmans 2006). However, in most cases, G ST, h, and u ST give highly similar values, so that their maximum values will also generally be close or equal to 1 ) H S. One statistic that is not affected by the amount of within-population variation is R ST, which was especially H S INVITED TECHNICAL REVIEW 7 developed for markers with a stepwise mutation model, such as some microsatellites (Slatkin 1995). Slatkin showed that estimates of the number of migrants, calculated from R ST, were essentially unbiased over a range of mutation rates and were much better than estimates calculated from F ST. Similar results were obtained by Balloux et al. (2000) who found that estimates of R ST were mostly unbiased for highly variable microsatellite loci with up to 30 alleles, while F ST was severely underestimated. However, estimates for R ST were only satisfactory when the mutations strictly followed the stepwise mutation model (Balloux et al. 2000). When a small proportion of random mutations were added, most of the memory in the mutation process was lost, and estimates of R ST were not reliable. As in practice microsatellites hardly ever follow a strict stepwise mutation model, the use of R ST is best avoided (but see Excoffier & Hamilton 2003). In fact, even when mutation is strictly stepwise, R ST is not necessarily always a better estimator than F ST (Balloux & Goudet 2002). For example, this is the case when the timescale of interest is short and the influence of mutation is relatively small (Slatkin 1995). Difficulties in interpretation Obviously, the dependency of many F ST estimators on the level of diversity will cause difficulties in their interpretation. This will be especially the case when markers are compared that have different mutation rates or when species are compared with different effective population sizes. Comparisons of F-statistics within species can also be difficult when different parts of the distribution are compared that differ in diversity. For example, an invasive species may have a lower diversity in the invaded area than in the original distribution area, which may give problems when F ST is used to compare the population structure within the two areas. Indeed, several authors have remarked that their estimates of F ST did not conform to the expectations based on what was known about their study organism or that the estimates varied over loci with different mutation rates (e.g. Balloux et al. 2000; O Reilly et al. 2004; Carreras-Carbonell et al. 2006). In a comparison of two chromosomal races of the common shrew, Balloux et al. (2000) found that the estimates of genetic differentiation were much lower than expected based on earlier studies. For example, for one highly variable Y-chromosomal microsatellite, the value of h was only 0.19, even though no alleles were shared between the two races. Ten autosomal microsatellite loci showed an even lower overall h value of In contrast, a biallelic mtdna marker where the two alleles were shared by the two races had a h value of Balloux et al. (2000) then conducted simulations to show that indeed the estimates of h were strongly affected by the mutation rates of the loci. Therefore, they concluded that, despite the low h values, the two races were genetically strongly differentiated as a result of almost complete reproductive isolation. O Reilly et al. (2004) used 14 microsatellite loci to study the population structure of a marine fish, Walleye pollock. The loci varied dramatically in the number of alleles (6 43), resulting in expected heterozygosities ranging from 0.68 to They found that the estimates of h declined significantly with increasing heterozygosity, leading them to conclude that mutation rates of some microsatellite loci are sufficiently high to limit resolution of weak genetic structure (O Reilly et al. 2004). They attributed the observed correlation to size homoplasy, downplaying the effect of heterozygosity itself as they regarded the population structure too weak to be affected by this. However, also when the population structure is weak, F ST will be affected by the level of heterozygosity and this can therefore readily explain the observed correlation. The proposed solutions Hedrick s G ST Having noted earlier (Hedrick 1999) that the diversity restricts the possible range of G ST, Hedrick (2005) suggested standardizing G ST by the maximum value it can obtain given the observed within-population diversity. This method of standardization was inspired by Lewontin s (1964) measure of linkage disequilibrium D, which is the standard measure D, divided by the maximum possible value given the observed allele frequencies. Hedrick used the original (Nei 1973) definition of G ST as (H T -H S ) H T and found that its maximum value (G ST(max) ) is a function of the expected heterozygosity, H S, and the number of sampled populations k G STðmaxÞ ¼ ðk 1Þ ð 1 H SÞ k 1 þ H S Hedrick then defined the standardized G ST, which he called G ST, as (equation 4b in Hedrick 2005) G 0 ST ¼ G ST ¼ G STðk 1 þ H S Þ ð1þ G STðmaxÞ ðk 1Þð1 H S Þ When k is large, G ST(max) becomes equal to 1 ) H S, the same value that was obtained above (Charlesworth 1998; Hedrick 1999; Jost 2008). The standardization ensures that G ST has an upper limit of 1, which is reached when the populations have nonoverlapping sets of alleles or when all populations are fixed for a single allele (H S =0) and there are two or more different alleles over all populations. 8 INVITED TECHNICAL REVIEW Hedrick defined his standardized measure only for G ST, but the rationale is also applicable to other F ST analogues. Meirmans (2006) developed a method to estimate the standardized measure u ST based on an analysis of molecular variance (AMOVA, Excoffier et al. 1992). In a normal AMOVA, the summary statistic u ST is defined as a function of the between-population variance component r a 2 and the within-population variance component r b 2 : r2 a u ST ¼ r 2 a þ r2 b The maximum value of u ST given the amount of withinpopulation variation can then be found by maximizing the among-population variance r a 2.InanAMOVA, the variance components are calculated from a matrix of pairwise squared Euclidean distances between individuals. For a single locus, the maximum among-group variance can then be found by setting all distances between pairs of individuals from different populations to a value of one. Jost s D Jost (2008) argued that there are in fact two separate problems with G ST and developed a new framework for analysing population differentiation based on ecological diversity theory. The first problem recognized by Jost is the same one that we saw above, where H S puts a limit on the maximum possible differentiation. Jost argued that the additive partitioning that is used for G ST, where the total diversity is the sum of the within-population and among-population diversity, is inadequate to describe the among-population diversity. The second problem recognized by Jost is that the expected heterozygosity is an unsuitable metric for describing the diversity, leading to unintuitive results. For example, the heterozygosity does not scale linearly with an increase in diversity. Going from two equally frequent alleles to 20 equally frequent alleles does not give a tenfold change in heterozygosity, but a more moderate change from 0.5 to Changing from 20 to 200 equally frequent alleles gives even less change in heterozygosity from 0.95 to For some uses, this is a desirable property, for example because the heterozygosity very well fits our human interpretation of diversity where changes in small numbers (from 2 to 20) are often considered more important than changes in large numbers (from 20 to 200) (Hubalek 2000). However, this quality makes the heterozygosity less suitable for a statistical breakdown of diversity. One additional advantage of using heterozygosity is its easy interpretation, because it presents the probability that a pair of randomly drawn genes are different. Jost (2008) then developed a new framework for estimating genetic differentiation that avoids these two problems. Instead of using heterozygosity, Jost based his statistic D on the effective number of alleles, which Jost (2006, 2008) simply calls the true diversity. The effective number of alleles scales linearly with an increase in equally frequent alleles, which, according to Jost, gives a more intuitive diversity estimate. A disadvantage of this diversity index is that it depends on the sample size, so rarefaction to a standard sample size is needed before estimates can be compared (note that this does not affect the estimation of D). The effective number of alleles is directly related to the heterozygosity and can be defined as 1 (1 ) H S ). However, unlike the heterozygosity, the effective number of alleles does scale linearly with increases in diversity. Jost (2007, 2008) then developed a multiplicative approach to partition the diversity, where the total diversity is the product of the within-population and among-population diversity and shows that this approach is mathematically more robust than the additive partitioning used by Nei. He then transformed the among-population diversity into a summary statistic, D, which ranges from zero to one. Although this statistic is not directly based on the heterozygosity as an index of diversity, it can nevertheless be expressed as a function of the total and within-population heterozygosities (equation 11 in Jost 2008): k HT H S D ¼ ð2þ k 1 1 H S If H S = 0, then D = H T k (k ) 1). This means that when k =2, D equals 2H T and when k becomes large, D approaches H T. Relationships between G ST,G ST and D To visualize the relationships between the three summary statistics, G ST, G ST and D, Heller & Siegismund (2009) collected data on 43 species from 34 studies published in Molecular Ecology between 2006 and They included all species for which estimates were given for both H S and an F ST analogue and used these estimates to calculate (by approximation) G ST, H T, D and G ST.We extended their data set and added another 41 species from 36 studies published in Molecular Ecology between January 2009 and March The strong positive correlation between G ST and D that was reported by Heller & Siegismund (2009) was also present in the extended data set, though with a slightly lower value for the correlation coefficient (r = 0.85 for the extended data set, vs. r = 0.99 for the smaller set). As G ST, G ST and D can all be expressed in terms of H S, H T and k, it is possible to directly analyse the relationships among these three statistics (Heller & Siegismund 2009). The relationship between G ST and G ST is simple INVITED TECHNICAL REVIEW 9 (a) G ST G ST (b) D G ST (c) D G ST H S H S H S Fig. 2 The relationships between G ST, G ST and D, as a function of the amount of within-population diversity H S. (a) The ratio between G ST and G ST, (b) the ratio between D and G ST, (c) the ratio between D and G ST. The thick grey lines outline the possible range of the expected relationship for a large number of populations, the thin dotted lines outline the possible range for k = 2. These upper and lower limits, respectively, assume that H T = 1 and H T = H S. The black symbols represent values from the literature (expanded from Siegismu
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