TY - JOUR
AU - De Meeûs, Thierry
AB - Abstract Null alleles and Wahlund effects are well known causes of heterozygote deficits in empirical population genetics studies as compared to Hardy–Weinberg genotypic expectations. Some authors have theoretically studied the relationship of Wright’s FIS computed from subsamples displaying a Wahlund effect and FST before the Wahlund effect, as can occasionally be obtained from populations of long-lived organisms. In the 2 subsample case, a positive relationship between these 2 parameters across loci would represent a signature of Wahlund effects. Nevertheless, for most organisms, getting 2 independent subsamples of the same cohort and population, one with a Wahlund effect and the other without, is almost never achieved and most of the time, empirical population geneticists only collect a single sample, with or without a Wahlund effect, or with or without null alleles. Another issue is that null allele increase FIS and FST altogether and thus may also create such correlation. In this article, I show that, for organisms collected in a single sample, which corresponds to the most common situation, Wahlund effects and null alleles affect the values of both FIS and FST though in the opposite direction. I also show that Wahlund effect produces no or weak positive correlation between the 2 F-statistics, while null alleles generate a strong positive correlation between them. Variation of these F-statistics is small and even minimized for FST under Wahlund effects as compared to null alleles. I finally propose a determination key to interpret data with heterozygote deficits. Population structure and phylogeography, differentiation, F-statistics, genetic identities, inbreeding Empirical population genetics is the study of the population biology of natural populations through the analysis of spatio-temporal variation of molecular markers (e.g., allozymes, microsatellite loci, SNPs, etc.). Hardy–Weinberg equilibrium and its expected genotypic proportions are central in such studies, because any deviation from these expected proportions can provide clues for inferences on the functioning of the targeted populations (De Meeûs et al. 2007). Many tools are currently available but some of the most widely used are Wright’s (Wright 1965) F-statistics (Nagylaki 1998): FIS, is a measure of deviation from panmixia at local scales (e.g., within subsamples); FST is a measure of subdivision; FIT is a measure of deviation from panmixia at the whole sample scale and actually corresponds to the combined effect of the 2 first ones: (1 − FIT) = (1 − FIS) (1 − FST) (De Meeûs et al. 2007). In a recent article, Waples (2015) reviews the different causes of deviations from Hardy–Weinberg genotypic expected proportions and linkage disequilibrium, with some attention, among other factors, to Wahlund effect and null alleles. Wahlund effect occurs when genotypic proportions are computed from heterogeneous samples where individuals belonging to genetically differentiated entities are pooled. These entities can represent members of different (and genetically distant) subpopulations, different cohorts, or different cryptic species. This typically happens when the real scale at which population structure occurs is unknown, resulting in an inaccurate sampling design. Null alleles occurs when the genotyping technique used fails to unveil the presence of true alleles that thus appear as missing data if homozygous or as apparent homozygous when heterozygous with a visible allele. This typically arises in microsatellite markers when a mutation event alters one of the flanking sequences where primers are to attach to initiate DNA amplification (PCR). If this mutation prevents the primer annealing to template DNA during amplification of the microsatellite locus by PCR, this results in a null allele. Wahlund effect and null alleles both increase FIS and thus produce apparent heterozygote deficits. In his article, Waples (2015) offered a criterion to discriminate a Wahlund effect from other causes as null alleles, in a 2 populations and 2 alleles context, which was generalized by Zhivotovsky (2015) for multiple alleles. In these models, the FST before Wahlund effect must be known from an independent sample and FIS measured in another sample with Wahlund effect. I will label the Wahlund effect on FIS as FIS_W. The main result of these works is that there is an expected positive correlation between FIS_W and FST across loci, which can be used as a criterion as opposed to null alleles where such correlation is believed not to happen. It was also stated that because of this correlation, the criterion that is often found in the literature of stability of FIS across loci under a Wahlund effect, as compared to the effect of null alleles, is void. It is important to specify that in these models, the true FST (without Wahlund effect) must be known and the regression undertaken between this true FST and the FIS is measured after the Wahlund effect (FIS_W). Such correlation between FST and FIS_W can also be easily (and with more generality) derived from the famous Wright’s equation (1 − FIT) = (1 − FIS) (1 − FST) because in a sample with a Wahlund effect, and by definition of those F-statistics, what is measured as FIS_W is necessarily strongly connected to the initial (i.e., “true”) FIT and is equal to it when the final sample is a random collection of existing real subpopulations. The knowledge of true FST is a rare situation that I only know of from a few studies on fairly long-lived mammal species: the Leadbeater’s possum in Australia (Waples 2015), and the North Pacific minke whale (Waples 2011). Long-lived organisms indeed allow sampling the same cohort after several years or months so that each locus keeps the same characteristic over time between 2 sampling campaign, one of which displays a Wahlund effect, and the other displays the true population subdivision. For most organisms, in particular small ones for which population genetics tools represent the only mean to study their population biology (De Meeûs et al. 2007), initial (true), FST and Wahlund FIS_W resulting from inaccurate sampling cannot be known together: either the sampling is accurate and FIS is not affected by a Walund effect, or sampling is not accurate and FIS and FST are both affected. Because of short generation times, even if 2 samples are available, one of which with true FST and the other with a Wahlund effect (a case that I have never met except for the 2 examples cited above), the correlation would have little chance to survive the redistribution of per locus statistics with genetic drift. A Wahlund effect typically occurs for organisms for which population subdivision is unknown, if subsamples are collected at larger scales than the actual subdivision unit. In that case, sampling contains a Wahlund effect that affects all statistics all together. Several such examples and discussions can be found in Ravel et al. (2007), Bouyer et al. (2009), Kempf et al. (2010), Prugnolle and De Meeûs (2010), Solano et al. (2010), and Rougeron et al. (2015). It is also known that null alleles also increase FST (Chapuis and Estoup 2007). A positive correlation between FIS and FST can thus be predicted in that case. This thus can lead to confusion. If both FST and FIS observed in the same sample are regressed and a positive correlation found, concluding to a Wahlund effect signature, as can be seen in some studies (Criscione et al. 2011; Bohling et al. 2016), may be wrong. In “classic” (i.e., single sample) samples, it can also be suspected that Wahlund effect also affects FST computations. In this article, I analyze the most common situation where F-statistics are estimated from the same sample. I show that null alleles increase both FIS and FST and that Wahlund effect increases FIS but decreases FST. Simulations show that when sampling and genotyping are accurate (no Wahlund effect and no null alleles), a negative correlation is expected between FIS and FST most of the time and that a Wahlund effect will weakly change this tendency. I also show that a strong positive correlation links FIS and FST in presence of null alleles with a maximal variation of these statistics across loci in that case, while Wahlund effect weakly (if any) affects FIS variation and minimize FST variance across loci. I finally propose a determination key to interpret data with heterozygote deficits. Because I am focusing on heterozygote deficits, I will not discuss selective processes. The only selective process that can cause heterozygote deficits is underdominance, a very unstable situation that should be met extremely rarely (De Meeûs et al. 2007). The only possible example could be the African butterfly Pseudacraea eurytus. But I could only find it in Wikipedia. A more convincing example corresponds to the rhesus system in human populations (De Meeûs et al. 2007; Abbey et al. 2011). Multilocus underdominance might happen in hybrid zones. Nevertheless, gathering different cryptic species (or subspecies) into single subsamples will more correspond to more or less extreme “classic” Wahlund effects, as treated in the present article. Methods Definitions In this article, I will use F-statistics definitions according to Cockerham (1969, 1973) and Rousset (1996, 2004) and a notation that I find easier to follow as in De Meeûs et al. (2007). In a 3 hierarchical (i.e., nested) level of sampling structure, with individuals in subsamples and total sample, we can define 3 identity parameters: QI is the probability to twice sample the same allele in an individual; QS is the probability to twice sample the same allele from 2 individuals from the same subsample; QT is the probability to twice sample the same allele from 2 different subsamples. From there, we can define Wrights F-statistics: {FIS=QI−QS1−QSFST=QS−QT1−QTFIT=QI−QT1−QT (1) Note that 1 − QS and 1 − QT are the probabilities to sample different alleles in one subsample or in the total sample, respectively, and are thus equal or close to subsample and total sample genetic diversities HS and HT, respectively. I will use the above notation for the “true” parameters, that is, the ones that would be measured if the population investigated would have been accurately sampled (no Wahlund effect) and the marker used immune from genotyping errors (no null alleles, dropouts, or stuttering). QI_W, QS_W, and QT_W are the probabilities of identity measured in subsamples with Wahlund effects and FIS_W, FST_W, and FIT_W are the corresponding F-statistics. WS and WT parameters describe the intensity of Wahlund effect on QS and QT, respectively. These parameters are a function of real allele frequencies in the different subpopulations and of the importance and modality of admixture of individuals from different subpopulations in the actual sample. These parameters are thus functions of FST. QI_N, QS_N, QT_N, FIS_N, FST_N, and FIT_N are the same as above but when the genetic marker is affected by null alleles. QIi, QSi, and FISi are the true corresponding parameter values in subpopulation i. In the bi-allelic case, pi and qi are the “true” allele frequencies of the 2 visible alleles of the genetic marker in subpopulation i and pNi is the frequency of null alleles in subpopulation i. The bar above a parameter will mean the average value across all subsamples: for instance, the average null allele frequency across n subsamples will be: pN¯=1n∑i=1npNi (2) Two population structure models, with n subpopulations of size N and immigration rate m were explored: the Island model, where each subpopulation, at each generation, is composed of (1 − m)N residents and mN immigrants coming from any of the n − 1 other subpopulations (no spatial structure); the 2 dimensional stepping stone model, where immigrants come from direct neighbors (4 neighbors in central subpopulations, 2 in corner subpopulations, and 3 for other marginal subpopulations). Stepping stone models thus result into isolation by distance (spatial population structure). Different sampling errors lead to a Wahlund effect. A Wahlund effect can be spatial and/or temporal. Spatial Wahlund effect can itself result from several kinds of sampling errors, some of which are illustrated in Figure 1. Each kind gathers individuals that belong to different subpopulations (with different allele frequencies). For the unshared Wahlund effect, each subsample contains individuals from several subpopulations that are not shared with other subsamples (Figure 1D). For the shared Wahlund effect, each subsample shares common origins with other subsamples with respect to subpopulations that are pooled (Figure 1B,F). A third sampling gathers heterogeneous admixtures (unbalanced Wahlund effect, Figure 1E). Wahlund effects arise from a lack of knowledge of the true structure of a population. Hence, “shared” and/or “unbalanced” samples are probably the most frequent causes of Wahlund effects. For populations with isolation-by-distance migration models (2-dimensional stepping stone models), the closest subpopulations are genetically the most similar. For Island models of migration with no spatial structure of genetic information, genetic differentiation is uncorrelated with subpopulation position. Figure 1. View largeDownload slide Different sampling designs that can lead to a Wahlund effect for simulations with 25 (A, B, D, E) or 64 subpopulations (C, F). Subsamples are represented in dark grey. Correct sampling (A, C) illustrates how subpopulations were initially sampled. For unshared Wahlund effect (D), subsamples gather individuals from different subpopulations but each subsample does not share individuals from the same subpopulation with other subsamples. In shared Wahlund effect (B, F), subsamples gather individuals from different subpopulations that are also shared between subsamples. For this case, subsamples 1 and 9 contain unbalanced representation of 2 subpopulations. In unbalanced Wahlund effect (E), the variance of representation of each initial subpopulation is maximized as compared to shared Wahlund effect (B, F). Figure 1. View largeDownload slide Different sampling designs that can lead to a Wahlund effect for simulations with 25 (A, B, D, E) or 64 subpopulations (C, F). Subsamples are represented in dark grey. Correct sampling (A, C) illustrates how subpopulations were initially sampled. For unshared Wahlund effect (D), subsamples gather individuals from different subpopulations but each subsample does not share individuals from the same subpopulation with other subsamples. In shared Wahlund effect (B, F), subsamples gather individuals from different subpopulations that are also shared between subsamples. For this case, subsamples 1 and 9 contain unbalanced representation of 2 subpopulations. In unbalanced Wahlund effect (E), the variance of representation of each initial subpopulation is maximized as compared to shared Wahlund effect (B, F). Temporal effects occur when members of different cohorts are pooled in the same subsamples. This is a classical cause of biased parameter estimates and statistics in microorganisms for which strains from different years are often pooled, most of the time to increase subsample sizes. Nevertheless, the gain in size is made at the expense of accuracy and power. Interested readers can find some more specific comments about population genetics of Leishmania parasites in Rougeron et al. (2015). I will only treat temporal effects in the discussion in the light of spatial Wahlund effects results. Simulations To check for analytical and simplification accuracy, I have undertaken several simulations. All simulations were undertaken with Easypop 2.0.1 (Balloux 2001) and analyzed with Fstat 2.9.4, updated from Goudet (1995). All simulations consisted of 25 or 64 subpopulations of 200 or 100 individuals with balanced sex ratio in Island or 2-dimensional stepping stone models with migration rates of m = 0.01, 0.05, 0.2, or 0.5, for 10000 discrete generations, 20 loci with a KAM mutation rate of u = 10−4 and K = 99 possible alleles. In some instances, I also simulated heterogeneous loci with various mutation rates and K: Loci 1–5, u = 10−9, K = 2; Loci 6–10, u = 10−6, K = 10; Loci 11–15, u = 10−5, K = 20; Loci 16–20, u = 10−4, K = 40. Hermaphrodites with a selfing rate of s = 0.5 were also simulated for several parameter sets, some of which (the most significant ones) are presented in the Results section. Each parameter set was replicated 10 times. For each replicate, I sampled 20 individuals in 10 subpopulations (Figure 1). Null alleles were generated as in Séré et al. (2014, 2017). For each locus, alleles 1–10 (10% of null alleles), 1–20 (20%), and 1–50 (50%) were recoded as null alleles. Hence, individuals harboring one of these alleles were recoded as homozygous for the other allele harbored, individuals harboring 2 of these alleles were recoded as missing data and individuals harboring none of these alleles were not recoded. At the end of each simulation, some alleles are lost by drift. This way, the same process (mutation and drift) affects all alleles (that are all present at generation 1) at all loci homogeneously, resulting in (approximately) 10%, 20%, or 50% of null allelic states among the K possible ones. All alleles (including null alleles) are then submitted to genetic drift and mutation like any other locus. This produces a large variation in null allele frequencies across loci and replicates (absence, weak, medium, or substantial frequency) as it is expected in real data with null alleles (De Meeûs et al. 2007; Séré et al. 2014, 2017). These proportions of null alleles (10%, 20%, and 50%) actually correspond to the proportions of allelic states that are null and not the actual real null allele frequencies which vary considerably across subpopulations and across replicates. For the sake of simplification, I will call these “proportion of null alleles.” Wahlund effects were produced as in Figure 1. Merging adjacent subpopulations (e.g., 6 with 7, 8 with 9, etc.) resulted in 5 subsamples with unshared Wahlund effect (see definitions and Figure 1D). In that case, all subsamples contain a balanced admixture of 2 subpopulations. For Shared Wahlund effect, subsample 1 (subsample S1, Figure 1) is built with 20 individuals from subpopulation 6 and 10 individuals from subpopulation 7 (for n = 25, Figure 1B) or, for n = 64 (Figure 1F), with 20 individuals from subpopulation 10 and 10 individuals from subpopulation 11. Subsequent subsamples are then built with 10 remaining individuals from the previous subpopulation and the 10 first of the following one. The last subsample (subsample S9, Figure 1B,F) is finally composed of 10 individuals from subpopulation 18 (Figure 1B) or 20 (Figure 1F) and 20 individuals from subpopulation 19 (Figure 1B) or 21 (Figure 1F). Here, the first and last subsamples contain unbalanced admixtures of 2 subpopulations while others contain balanced admixtures. Finally, the unbalanced Wahlund effect was generated through heterogeneous admixtures in subsamples (Figure 1E). For stepping-stone models, Wahlund effects are expected weaker than for Island models since for the first, merged subpopulations are neighbors and hence are genetically relatively closed even if m is small. For Island model, merged subpopulations are genetically similar or distant, depending on m. Spearman’s rank correlation test was undertaken with the package Rcmdr (Fox 2005, 2007) for R 3.3.2 (R Development Core Team 2016) and 95% confidence intervals for different statistics were estimated using the variation of these statistics across replicates of the same simulation. When indicated, 95% confidence intervals of FIS and FST were computed using the standard error of jackknife over subsamples (for each locus) or using 5000 bootstraps over loci (for the average across loci). These were computed with Fstat. More detailed explanations on how to compute confidence intervals can be found in De Meeûs et al. (2007). The correlation between FST and FIS across loci was measured with Pearson’s correlation coefficient. Results Wahlund Effects In the case of a Wahlund effect, the total sample is composed of several subsamples that contain admixtures of individuals from different real subpopulations in proportions that are unknown and variable from one subsample to the other. We can write identity probabilities as (Equation 1): {QI_W=QI QS_W=QS−WSQT_W=QT+WT (3) There is indeed no reason why a Wahlund effect would affect the probability of homozygosity QI. On the contrary, the presence of new alleles from different subpopulations will tend to decrease the probability of identity between individuals in each subsample, and hence 0 ≤ WS ≤ QS. WS will more depend on the level of differentiation between gathered subsamples (FST) than on sampling design itself. The value of WT will depend more on the sampling design. In “unshared subsamples,” its value should be close to 0. In “shared” or “unbalanced” subsamples, the probability to sample the same allele from different subsamples should increase more, but the difference might be very small in fact. Hence, 0 ≤ WT ≤ 1 − QT. High values should not be frequently met in real situations. Temporal Wahlund effects will be similar though less predictable, in particular, on WT. With very heterogeneous cohort composition from one site to the other, it might happen that WT < 0, particularly so in short generation time species with small effective population sizes (e.g., microbes) that can experience swift divergences between different cohorts. This might be clearer with a caricatured example. If subsample 1 at site 1 is composed of cohorts 1 and 4, and subsample 2 at site 2 is composed of cohorts 7 and 10, in a short lived species with small effective population sizes, drift will be quick and genetic differentiation between cohorts 4 and 7 might be bigger than between site 1 cohort 1 and site 2 cohort 1. Nevertheless, this last situation probably occurs only rarely. Under a Wahlund effect, and combining Equation 1 and 3, FIS and FST become: {FIS_W=QI−QS+WS1−QS+WS FST_W=QS−WS−QT−WT1−QT−WT (4) It can easily be shown that, whatever initial identity probabilities, FIS_W ≥ FIS and FST_W ≤ FST. Please, note that, as seen above, special temporal Wahlund effects might end up with FST_W > FST. This situation is probably exceptional because it requires that WT < 0 and |WT| > |WS|. We can rearrange Equation 4 into: {FIS_W[1−QS+WS]=QI−QS+WS FST_W=QS−WS−QT−WT1−QT−WT ⇔{−(QI−QS)+FIS_W(1−QS)=WS[1−FIS_W]FST_W=QS−WS−QT−WT1−QT−WT ⇔{WS=FIS_W(1−QS)−(QI−QS)1−FIS_W FST_W=QS−FIS_W(1−QS)−(QI−QS)1−FIS_W−QT−WT1−QT−WT⇔{WS=FIS_W(1−QS)−(QI−QS)1−FIS_W FST_W=QS−QT−WT1−QT−WT+QI−QS1−FIS_W×11−QT−WT−FIS_W1−FIS_W×1−QS1−QT−WT (5) In most situations we expect WT to be small so that we could rewrite Equation 5 as: {WS=FIS_W(1−QS)−(QI−QS)1−FIS_W FST_W≈QS−QT1−QT+QI−QS1−FIS_W×11−QT−FIS_W1−FIS_W×1−QS1−QT⇔{WS=FIS_W(1−QS)−(QI−QS)1−FIS_W FST_W≈FST+QI−QS1−FIS_W×11−QT−FIS_W1−FIS_W×1−QS1−QT (6) If we assume local panmixia (QS = QI). In that case Equation 6 becomes: {WS=FIS_W1−FIS_W(1−QI) FST_W≈FST−FIS_W1−FIS_W×1−QI1−QT or {WS=FIS_W1−FIS_WHI FST_W≈FST−FIS_W1−FIS_W×HIHT (7) where HI is the proportion of heterozygotes in the total sample (or its average across subsamples) and HT is the total genetic diversity of the sample, which is approximately the same as in the real population if WT is small. Equation 7 can be rearranged into: {WS=FIS_W1−FIS_WHI FST≈FST_W+FIS_W1−FIS_W×HIHT (8) which provides a rough proxy of the true FST, assuming local panmixia and small impact of the Wahlund effects on total genetic identity. Please note that this proxy will be very bad if the subpopulations are not panmictic and the impact of the Wahlund effect on QT is substantial. It might seem from Equation 7 that FIS_W and FST_W are negatively correlated but this is compensated by the fact that FIS_W is more strongly correlated to FST (and positively so) than to FST_W (negatively so). Null Alleles This case is in fact more difficult to track analytically. I will thus directly assume panmixia in each subsample with 3 alleles (1, 2, and null). In that case the (unseen) real genotypic proportions in subsample i are pi2, 2piqi, qi2, 2pipNi, 2qipNi, and pNi2 for genotypes 11, 12, 22, 1N, 2N, and NN, respectively. We can also express QIi and QSi: QIi=QSi=pi2+qi2+pNi2 (9) We can now compute the perceived allele frequency of allele 1 in subsample i, remembering that 1N and 2N individuals are erroneously interpreted as homozygous 11 and 22, respectively: pi_N=pi2+2pipNi+12×2piqi1−pNi2⇔pi_N=pipi+2pNi+qi1−pNi2=pi1+pNi1−pNi2=pi1+pNi(1−pNi)(1+pNi)⇔{pi_N=pi(1−pNi)qi_N=qi(1−pNi) (10) From there, the observed within individual genetic identity in subsample i with null alleles can be written: QI_Ni=pi2+qi2+2pNi(pi+qi)1−pNi2 (11) We can rearrange Equation 11 and combine it with Equation 9: QI_Ni=pi2+qi2+2pNi(pi+qi)1−pNi2 (12) To understand the possible values taken by QI_Ni, it is useful to compute the minimum and maximum possible values for QS, with a null allele frequency pNi. Let us call these quantities QSi_min/N and QSi_max/N. Probability of identity between alleles 1 and 2 will be minimal when both share the same allele frequency (1 − PNi)/2 and will be maximal when one is very close to 0 and the other with frequency close to 1 − pNi. We can thus write: {QSi_min/N=2×[12(1−pNi)]2+pNi2QSi_max/N=(1−pNi)2+pNi2 ⇔{QSi_min/N=12(1−pNi)2+pNi2QSi_max/N=(1−pNi)2+pNi2 (13) We can use this to check the range of variation of QI_Ni for a given pNi combining Equations 11 and 13. Its extremum values are then: {QI_Ni_min/N=12(1−pNi)2+pNi2−pNi2+2pNi(1−pNi)1−pNi2QI_Ni_max/N=(1−pNi)2+pNi2−pNi2+2pNi(1−pNi)1−pNi2 ⇔{QI_Ni_min/N=(1−pNi)12(1−pNi)+2pNi(1−pNi)(1+pNi)QI_Ni_max/N=(1−pNi)(1−pNi)+2pNi(1−pNi)(1+pNi) Since pNi > 0 we can write: {QI_Ni_min/N=12(1+3pNi1+pNi)QI_Ni_max/N=1+pNi1+pNi ⇔{QI_Ni_min/N=12(1+2pNi1+pNi)QI_Ni_max/N=1 (14) It can easily be seen that the upper term of Equation 14 is minimized when null allele frequencies are very small (and their effect negligible) and then equals ½ (as expected for the bi-allelic case). We can thus write ½ ≤ QI_Ni ≤ 1. We now need to derive equations for QS_Ni. Combining Equations 9 and 10, we obtain QS_Ni=(pi1−pNi)2+(qi1−pNi)2=1(1−pNi)2(pi2+qi2) (15) In subsample i, combining Equations 1, 12, and 15, FIS_Ni will have the same sign as: ΔQIS_Ni=QI_Ni−QS_Ni=QSi−pNi2+2pNi(1−pNi)1−pNi2−1(1−pNi)2(QSi−pNi2)⇔ΔQIS_Ni=(1−pNi)[QSi−pNi2+2pNi(1−pNi)]−(1+pNi)(QSi−pNi2)(1+pNi)(1−pNi)2⇔ΔQIS_Ni=2pNipNi2+(1−pNi)2−QSi(1+pNi)(1−pNi)2⇔ΔQIS_Ni=2pNipNi2+(1−pNi)2−QSi(1+pNi)(1−pNi)2⇔ΔQIS_Ni=2pNi1−[2pNi(1−pNi)+QSi](1+pNi)(1−pNi)2 (16) This quantity is always positive (or null when QI_Ni = QS_Ni = 1, no polymorphism), because QSi is the sum of true homozygote frequencies and hence Equation 16 can be rearranged as: ΔQIS_Ni=2pNi1−(1−2piqi)(1+pNi)(1−pNi)2⇔ΔQIS_Ni=2pNi2piqi(1+pNi)(1−pNi)2 As expected, this quantity is always positive or null and then FIS_Ni ≥ 0. Over the n subsamples: {QI_N¯=1n∑i=1nQI_NiQS_N¯=1n∑i=1nQS_Ni (17) and FIS_N¯=QI_N¯−QS_N¯1−QS_N (20) If some polymorphism is maintained in some of the subsamples, then the quantity defined in Equation 20 is always positive. The real (unseen) total identity probability can be computed as: QT=p¯2+q˙¯2+pN¯2 (21) The seen total genetic identity, with null alleles at frequency pNi in subsample i, and combining Equations 10 and 21, can be written as: QT_N=(1n∑i=1npi1−pNi)2+(1n∑i=1nqi1−pNi)2⇔QT_N=1n2[(∑i=1npi1−pNi)2+(∑i=1nqi1−pNi)2] From there some numerical explorations can lead to simple rules that will simplify the algebra. QS≤QT {QI_N=QI+νI QS_N=QS+νS QT_N=QT+νT where 0 ≤ νT ≤ νS < νI ≤ 1 − QI_N. Indeed, the effect of null alleles will always be much greater on homozygosity than on identity between individuals and the effect of null alleles on identity between individuals will generally tend to be slightly higher than on identity between subsamples, especially so if the population is substantially subdivided. For instance if QT~0, then the effect of null alleles will be very weak (if any) on this parameter (νT~0), while QS = QT (no subdivision), necessarily means that νT = νS. We can thus write: {FIS_N=QI+νI−QS−νS1−QS−νS FST_N=QS+νS−QT−νT1−QT−νT (22) It is not difficult to show that FIS_N ≥ FIS and FST_N ≥ FST. Here, because both statistics are positively correlated to null allele frequencies, we may expect a positive correlation between FIS_N and FST_N. For more precisions on how to compute null allele frequencies I encourage readers to have a look at classic papers and reference therein (Brookfield 1996; Van Oosterhout et al. 2004; David et al. 2007; Guillot et al. 2008; Chybicki and Burczyk 2009; Robledo-Arnuncio and Gaggiotti 2017). Simulations With correct sampling and unaffected genotypes (no null alleles), we can see from Figure 2 that there is a strong (but not systematic) tendency for a negative correlation between FIS and FST across loci as measured by Pearson’s correlation coefficient. This tendency seems less pronounced for substantially inbred subpopulations (s = 0.5), small subpopulations (N = 100) and for heterogeneous loci (variable K and u). This is probably connected to the level of polymorphisms that can be maintained in subpopulations as illustrated by a highly significant negative correlation with HS (Figure 3). Figure 2. View largeDownload slide Average Pearson’s correlation coefficient [r(FIS, FST)] between FIS and FST across loci and 95% confidence intervals for simulations in Island or 2-dimensional stepping stones models (stepping stone) with different subpopulation sizes (N), number of subpopulations (n), migration rates (m), and different models of mutation: K = 99 for 99 alleles and mutation rate u = 10−4 and VarK for variable K and u as defined in the text. The results obtained for correct data (no Wahlund effect, no null alleles) (C), for data with unshared (U), shared (S), or unbalanced (Un) Wahlund effects, as defined in the text and with 50% selfing (s), different null allele frequencies (0.1, 0.2, 0.5) or random mating without null alleles (all others) are presented. Note that, for the sake of comparison, Wahlund, null alleles, or selfing always occur alone (no combination). The straight plain line in the middle represents the 0 correlation value. Figure 2. View largeDownload slide Average Pearson’s correlation coefficient [r(FIS, FST)] between FIS and FST across loci and 95% confidence intervals for simulations in Island or 2-dimensional stepping stones models (stepping stone) with different subpopulation sizes (N), number of subpopulations (n), migration rates (m), and different models of mutation: K = 99 for 99 alleles and mutation rate u = 10−4 and VarK for variable K and u as defined in the text. The results obtained for correct data (no Wahlund effect, no null alleles) (C), for data with unshared (U), shared (S), or unbalanced (Un) Wahlund effects, as defined in the text and with 50% selfing (s), different null allele frequencies (0.1, 0.2, 0.5) or random mating without null alleles (all others) are presented. Note that, for the sake of comparison, Wahlund, null alleles, or selfing always occur alone (no combination). The straight plain line in the middle represents the 0 correlation value. Figure 3. View largeDownload slide Relationship between r(FIS, FST) (Pearson’s correlation between FIS and FST) and local genetic diversity (HS) averaged across 10 replicates for each parameter sets with correct data (no Wahlund effect, no null alleles) as described in the text. Regressed 95% confidence intervals computed from variation across replicates are indicated as dotted lines. The regression determination coefficient (R2), Sperman’s rho and its corresponding P-value are also provided. The straight plain line in the middle represents the 0 correlation line. Figure 3. View largeDownload slide Relationship between r(FIS, FST) (Pearson’s correlation between FIS and FST) and local genetic diversity (HS) averaged across 10 replicates for each parameter sets with correct data (no Wahlund effect, no null alleles) as described in the text. Regressed 95% confidence intervals computed from variation across replicates are indicated as dotted lines. The regression determination coefficient (R2), Sperman’s rho and its corresponding P-value are also provided. The straight plain line in the middle represents the 0 correlation line. It can be seen from Figure 2 that Wahlund effects weakly increases the average correlation between FIS and FST, which stays below 0 most of the time, except in strongly subdivided populations. Even in this case, this correlation hardly exceeds 0.1 and displays an important variance. As expected with null alleles, correlation between FIS and FST is always positive (Figure 2), except in weakly subdivided populations with high null allele frequencies (pN = 0.5) (Supplementary Figure S1). Nevertheless, if loci with no heterozygosity (HI = 0) are removed from the data, all simulations with null alleles showed a strong positive correlation between the 2 statistics, especially so for strongly subdivided populations (Figure 2). Figure 2 presents a subset of all simulations undertaken with the most significant results. Other results can be found in the Supplementary File S1. The variation of FIS and FST across loci was measured with the standard error of jacknife over loci as computed in Fstat (StrdErrFIS and StrdErrFST, respectively). The results are presented in Figure 4. It can be seen that FST variation across loci tends to decrease from correct sampling (no Wahlund) to unshared Wahlund effect and shared Wahlund effect while it remains low and unchanged for FIS (no effect of sampling). A higher impact can be observed with loci presenting heterogeneous mutation models or smaller subpopulations (N = 100). For both statistics, for 10% of null alleles, the variation across loci jumps to much higher values. This is particularly true for FIS for which the standard error obtained with null alleles has nothing in common with simulations without null alleles of any kind. When there is no Wahlund effect, it can be seen that StrdErrFIS ≈ StrdErrFST, including simulations with selfing s = 0.5. In other situations, we can see that StrdErrFIS > StrdErrFST (Wahlund effects) or StrdErrFIS >> StrdErrFST (null alleles). Figure 4. View largeDownload slide Standard error of FIS (grey) and FST (black) from jacknife over loci as computed by Fstat and 95% confidence intervals. The results presented corresponds to 2-dimensional stepping stone or Island models with different subpopulation sizes (N), number of subpopulations (n), migration rates (m), homogeneous mutation models (with 99 alleles and 10−4 mutation rate) (K = 99), heterogeneous mutation models (VarK) and different situations: Correct sampling and genotyping (C), unshared (U), Shared (S), or unbalanced (U) Wahlund effects, 10% of null alleles (0.1) and 50% selfing rate (s). Note that, for the sake of comparison, Wahlund, null alleles, or selfing always occur alone (no combination). Figure 4. View largeDownload slide Standard error of FIS (grey) and FST (black) from jacknife over loci as computed by Fstat and 95% confidence intervals. The results presented corresponds to 2-dimensional stepping stone or Island models with different subpopulation sizes (N), number of subpopulations (n), migration rates (m), homogeneous mutation models (with 99 alleles and 10−4 mutation rate) (K = 99), heterogeneous mutation models (VarK) and different situations: Correct sampling and genotyping (C), unshared (U), Shared (S), or unbalanced (U) Wahlund effects, 10% of null alleles (0.1) and 50% selfing rate (s). Note that, for the sake of comparison, Wahlund, null alleles, or selfing always occur alone (no combination). Discussion The tendency for negative correlation across loci that links FIS and FST under the null hypothesis (no Wahlund and no null alleles) was unexpected (at least by me). It probably comes from the dependency, in an opposite direction, of both statistics on QS. Nevertheless, this correlation displays a strong variance around the average. There is thus little hope that this can often offer a useful criterion, especially when polymorphism is weak (HS<0.5). A strong negative correlation, more likely when HS > 0.6, will however represent a strong argument against the presence of null alleles. Contrary to what might be understood from the literature (Criscione et al. 2011; Waples 2015; Zhivotovsky 2015; Bohling et al. 2016), the correlation across loci between FIS and FST under a Wahlund effect will be very weak and most of the time even negative, while this correlation is always positive in the presence of null alleles, except in extreme situations with very high null allele frequencies, weak subdivision and if extreme loci displaying very low heterozygote frequencies are kept. The variation across loci for F-statistics does not tend to increase with Wahlund effect but at best remains unchanged or even decreases. On the contrary, with null alleles, this variation significantly increases, even for small null allele proportions (Figure 4). For FIS, variation across loci remains strictly the same under Wahlund effects except when the mutation model varies considerably across loci or in smallest subpopulations. For FST, even selfing generates more variation than Wahlund effect (Figure 4). But again, these variations are very modest as compared to those generated by null alleles. Temporal Wahlund effect should have similar consequences as geographic Wahlund effect though it may affect total genetic identity more strongly, depending on the cohort composition of each subsample and drift speed (hence population structure). In case of highly heterogeneous cohort compositions across subsamples, an increase of perceived FST might happen, but this should not be the most frequent situation. Different phenomena can affect heterozygosity all together in a single sample. Selfing and/or null alleles and/or Wahlund effect will then have different contribution that will be difficult to identify specifically. The classic Wright’s equation (1 − FIT) = (1 − FIS) (1 − FST) (e.g., De Meeûs et al. 2007) gives the relationship between the different F-statistics. FIT represents the combination of Wahlund effect (FST) and other causes (null alleles or deviation from panmixia) (FIS). The true FST can sometimes be known as described in Waples (2011, 2015) for long-lived mammals, in which case the true FIS might be extrapolated easily. Most of the time, such information is not available and adjusting data for the true FIS will be necessary. This can be made, for instance, by computing the contribution of null alleles to the FIS. A proxy can be found with the determination coefficient of the regression between missing genotypes counts and per locus FIS (if the correlation is good) (e.g., Melachio et al. 2011). Cross experiments can also be useful at determining possible range for selfing or sib mating rates, when such experiments are possible. Wahlund effect results from a poor knowledge of the actual structure of a given population. There is thus little chance that real FST and FIS can be known in real situations dealing with a Wahlund effect. As shown elsewhere (Chapuis and Estoup 2007) and in the present study, null alleles and Wahlund effects alter both FIS and FST, though in a different direction. Null alleles produce an increase of both statistics; while Wahlund effects increase FIS and, in most situations, decrease FST (FST might increase with temporal Wahlund effect with extreme heterogeneity of cohort composition). There will always be variation of F-statistics across loci, even under the null hypothesis, with some loci significantly deviating from panmictic proportions by chance only [for an excellent discussion on this issue see Waples (2015)]. This is why only a global significance across loci and subsamples represents the best clue that something is happening in the population under study (e.g., deviation from random mating, Wahlund effect, genotyping miscoring). In case of global significant deviation from Hardy–Weinberg proportions, excessive variation across loci of F-statistics represents an accurate signature of null alleles, or at least of locus-specific genotyping problems. For instance, it is known that allelic dropouts have similar effects as null alleles (Séré et al. 2014). On the contrary, Wahlund effects have no or very weak influence on this variation. Additionally, null alleles also produce strong positive correlations between FIS and FST, while Wahlund effects or selfing seldom do. To this respect, high variation of F-statistics across loci with a positive correlation between them will always represent a hallmark of locus-specific amplification problems (null alleles, dropouts, and stuttering). The absolute value of the jacknife standard error of FIS and FST may not be very useful as other factors as sample size and number of loci and their polymorphism may produce relatively high values. Nevertheless, plotting the variation of FIS and FST as in Figures 5 and 6 is always useful. These figures provide examples of what kind of variations can be attributed to null alleles or not (especially so for FIS). The presence of missing data (putative null homozygotes) can also helpfully add arguments in favor of null alleles, particularly so when the regression between FIS across loci as a function of the number of observed missing genotypes is significant (e.g., Melachio et al. 2011). Alternatively, relatively stable F-statistics with a correlation r(FIS, FST) around 0 or just above, and of course a significant positive FIS should strongly suggest a Wahlund effect, though this pattern will sometimes be hard to distinguish from the effect of local deviation from panmixia (selfing or sib-mating). Nevertheless, under ideal conditions, StrdErrFIS > StrdErrFST should correspond to Wahlund effects more than to selfing or sib-mating. I thus propose the following determination key for data with global (across loci and subsamples) significant heterozygote deficits: Figure 5. View largeDownload slide Example of the variation of FIS across loci, with 95% confidence intervals of jackknife over subsamples (each locus) or of bootstrap over loci (all) in different population models (Island or 2-dimensional stepping stone), one mutation model (K = 99, u = 10−4 for all loci), subpopulation numbers and size (n and N), with shared, unshared, or unbalanced Wahlund effects or with pN = 0.1 (proportion of null alleles). The straight plain line in the middle represents the 0 value. Figure 5. View largeDownload slide Example of the variation of FIS across loci, with 95% confidence intervals of jackknife over subsamples (each locus) or of bootstrap over loci (all) in different population models (Island or 2-dimensional stepping stone), one mutation model (K = 99, u = 10−4 for all loci), subpopulation numbers and size (n and N), with shared, unshared, or unbalanced Wahlund effects or with pN = 0.1 (proportion of null alleles). The straight plain line in the middle represents the 0 value. Figure 6. View largeDownload slide Example of the variation of FST across loci, with 95% confidence intervals of jackknife over subsamples (each locus) or of bootstrap over loci (all) in different population models (Island or 2-dimensional stepping stone), one mutation model (K = 99, u = 10−4 for all loci), subpopulation numbers and size (n and N), with shared, unshared, or unbalanced Wahlund effects or with pN = 0.1 (proportion of null alleles). The straight plain line in the middle represents the 0 value. Figure 6. View largeDownload slide Example of the variation of FST across loci, with 95% confidence intervals of jackknife over subsamples (each locus) or of bootstrap over loci (all) in different population models (Island or 2-dimensional stepping stone), one mutation model (K = 99, u = 10−4 for all loci), subpopulation numbers and size (n and N), with shared, unshared, or unbalanced Wahlund effects or with pN = 0.1 (proportion of null alleles). The straight plain line in the middle represents the 0 value. r(FIS, FST) >> 0 when excluding loci with HI ≈ 0 (if any), high variations of FIS and FST across loci, StrdErrFIS >> StrdErrFST and occurrence of missing data that explains FIS variation across loci: null alleles are affecting the data and the frequency of theses null alleles can be estimated with (for instance) MicroChecker (Van Oosterhout et al. 2004) and FST can be corrected with the ENA algorithm (Chapuis and Estoup 2007); additionally, if panmixia is never or rarely met for any locus, an additional phenomenon (Wahlund effect, selfing) might be invoked; r(FIS, FST)~0 or even moderately positive, very small variations of FST across loci, moderate variations of FIS across loci, StrdErrFIS > StrdErrFST and no or very rare missing data: Wahlund effect between subpopulations with an approximate FST given by Equation 8 (assuming local panmixia); r(FIS, FST) ≈ 0, moderate variance in FIS and FST across loci, StrdErrFIS = StrdErrFST and no or very rare missing data: selfing or sib mating might better explain the data, the rate of which can be estimated from FIS (e.g., De Meeûs 2012 p. 42 and 309). One additional criterion may help decision making. It uses the regression of per locus FIS (averaged over subsamples) as a function of missing genotypes observed in all subsamples. A significant positive relationship is a very sound indication that null alleles explain all or part of observed FIS. In that case, the determination coefficient (R2) will roughly indicate how much of variance of FIS across loci is explained by null alleles (Melachio et al. 2011). Data Availability All detailed simulation results, with corresponding graphics and simulation output files are available as Supplementary File S1. The genuine file can be downloaded from http://t-de-meeus.fr/Data/DeMeeusSimutationResultsNuls&WahlundSupFileS1.xlsx. Simulation results represent more than 1200 files. These files can easily be generated again with appropriate settings but the author is ready to share all or part of it with the community on request at thierry.demeeus@ird.fr. 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TI - Revisiting FIS, FST, Wahlund Effects, and Null Alleles
JF - Journal of Heredity
DO - 10.1093/jhered/esx106
DA - 2017-11-20
UR - https://www.deepdyve.com/lp/oxford-university-press/revisiting-fis-fst-wahlund-effects-and-null-alleles-cfg2iK1MqB
SP - 1
EP - 456
VL - Advance Article
IS - 4
DP - DeepDyve
ER -