Exegeses on Maximum Genetic Differentiation

In this commentary, François Rousset examines the topic of F_ST, a descriptor of spatial genetic structure. The commentary explores the issue of alternative descriptors, which are rejected in lieu of a new interpretation of F_ST in Jakobsson et al., “The Relationship Between F_ST and the Frequency of the Most Frequent Allele ”, published in the February issue of GENETICS.

DISTINCT neutral loci in a genome share a common distribution of genealogical relationships. However, in general these relationships cannot be directly inferred from genetic data, which additionally depend on marker features such as mutation rates and homoplasy (i.e., recurrent mutations to the same allelic type). Different conceptions of this discrepancy have led to sometimes conflicting definitions of F_ST and related concepts, such as identity by descent. A definition that is directly applicable to genetic data and that reflects the dependence on marker features can be given in terms of probability of identity in allelic state, Q, or equivalently in term of gene diversity, H = 1 − Q. Namely, F_ST can be defined as (Q_w − Q_b)/(1 − Q_b), in terms of the probabilities of identity within subpopulations (Q_w) and between subpopulations (Q_b), or equivalently as (H_b − H_w)/H_b. According to this definition, F_ST values cannot be higher than the within-deme identity, Q_w, which itself depends on mutational features intrinsic to the genetic markers. Interest in F_ST stems in part from its relative robustness with respect to these confounding marker-specific factors (e.g., Crow and Aoki 1984), but this robustness is far from perfect.

F_ST can be redefined so that there is no such dependence, as C_ST = (T_w − T_b)/T_b in terms of the average coalescence times of genes within subpopulations (T_w) and between subpopulations (T_b). This definition is appropriate for generalizing Wright’s insights (Slatkin 1991), but the traditional estimators of F_ST are not estimators of C_ST insofar as they are affected by mutation. This can be addressed in several nonexclusive ways, such as using F_ST only in conditions where it is expected to closely approximate C_ST or performing likelihood analyses that do not necessarily consider F_ST or C_ST as model parameters, but can nevertheless provide estimates of them as a byproduct.

However, the fact that F_ST cannot always reach 1 independently of Q_w is often perceived as a deficiency, independently of its relationship with C_ST. A possible correction is then to divide F_ST by its maximum value given Q_w, which is taken to be Q_w itself (Hedrick 2005). However, it is not clear what such a corrected F_ST (“$G_{\text{ST}}^{\prime }$”) brings. What one expects from the distribution of a statistic is that it is sensitive to parameters of interest and insensitive to other parameters, but there is no evidence that the corrected F_ST has such properties. On the contrary, the correction would bias inferences from F_ST in realistic conditions where the uncorrected F_ST gives robust answers despite low Q_w (e.g., Leblois et al. 2003; Whitlock 2011). Concerns about maximum F_ST values, and additional a priori arguments about what a differentiation measure should be, have also paved the way for the rejection of F_ST as not being a true measure of genetic differentiation (Jost 2008). The absence of a clear inferential framework raises the same questions for Jost’s proposed alternative as for $G_{\text{ST}}^{\prime }$ (Whitlock 2011). Thus, the proposed alternative descriptors deal with maximum values of the descriptors in a cosmetic way, which addresses none of the more substantial issues that make F_ST biases a concern.

In a recent paper in Genetics, Jakobsson et al. (2013) also reject these alternative descriptors and instead consider the relationship between the maximum F_ST value given the allele frequency of the most common allele. In essence, a common allele has to be present in different populations, and then F_ST, which depends on the variation in allele frequency among populations, cannot take large values. Detailed results demand some thought in the multiallelic case, and Jakobsson et al. (2013) further compute the expected maximum F_ST under some prior distribution for allele frequencies, not based on a population model. Taking examples from human populations, they emphasize that their calculations also show that highly polymorphic loci should have lower F_ST and can explain the relatively low estimates of F_ST for microsatellites. Actually, past works have recognized the latter effects at both theoretical and data analytical levels (e.g., Slatkin 1995; Rousset 1996; Estoup et al. 1998; Balloux et al. 2000). It has also been recognized that these effects contribute to the explanations for differences between single nucleotide polymorphisms (SNPs) and microsatellites in human populations (Payseur and Jing 2009).

Jakobsson et al. (2013) emphasize that their results for conditional F_ST hold independently of any biological process controlling allele frequency distributions, but a drawback of this model-free approach is that the biological conditions that make F_ST biases a concern are not identified. By comparison, earlier studies have addressed this question by taking expectations of gene diversities over distributions of population allele frequencies determined by different biological scenarios. In particular, Jakobsson et al. (2013) think that F_ST’s among human African populations “underpredict the intuitive level of differentiation” and that this can be explained in terms of the high genetic diversity of these populations. Although one cannot argue about intuitive levels of differentiation, model-based comparisons of F_ST to C_ST have shown that F_ST can approximate C_ST well despite high diversity, allowing low-bias inferences of demographic parameters from highly polymorphic markers (e.g., Figure 3 in Leblois et al. 2003). Whether F_ST’s of highly polymorphic markers will be biased or not can be understood by comparing the distributions of coalescence times of the different pairs of genes from which F_ST is defined (Rousset 1996).

Jakobsson et al. (2013)’s results concern maximum F_ST values, and a similar dependence of conditional average F_ST values can be expected. Such values are often considered independent from the allele frequency in the total population, at least when the total population comprises many local populations. However, Jakobsson et al. (2013) consider a pair of populations, and expected values of F_ST for such a pair, conditional on allele frequency in the same pair, will be lower than average when allele frequency is extreme in a biallelic system. Conversely, they will be higher than average when allele frequency is intermediate, with a variable strength of this dependence according to which frequency (from the pair or the total population) is taken as the conditioning variable (Rousset 2002). If an unbiased F_ST estimator is sought, averages over the distribution of the conditioning allele frequency should be computed, and it is not clear why we should look at conditional maximum or expected F_ST values. For other problems, such as the detection of selected markers, it may be relevant to consider the conditional distribution of an F_ST estimator, for example given the gene diversity (Beaumont and Nichols 1996).

In an inferential perspective, it is well known that the widely used F_ST estimators do not use all the information about model parameters in the data. With progress in likelihood methods for inference in population genetics, one could have expected a decreased interest in F_ST (e.g., Beerli and Felsenstein 2001). This has not occurred, and may not occur soon for reasons as diverse as authors’ fear of rejection for omitting some method, or the persistent difficulties of performing likelihood analyses. Thus, one can expect that statistical properties of F_ST and alternative summary statistics will remain a matter for debate, but such debates will be inconclusive as long as they rest only on model-free results.