Abstract
Studying isolation by distance can provide useful demographic information. To analyze isolation by distance from molecular data, one can use some kind of genetic distance or coalescent simulations. Molecular markers can often display technical caveats, such as PCRbased amplification failures (null alleles, allelic dropouts). These problems can alter population parameter inferences that can be extracted from molecular data. In this simulation study, we analyze the behavior of different genetic distances in Island (null hypothesis) and stepping stone models displaying varying neighborhood sizes. Impact of null alleles of increasing frequency is also studied. In stepping stone models without null alleles, the best statistic to detect isolation by distance in most situations is the chord distance D_{CSE}. Nevertheless, for markers with genetic diversities H_{S}<0.4–0.5, all statistics tend to display the same statistical power. Marginal subpopulations behave as smaller neighborhoods. Metapopulations composed of small subpopulation numbers thus display smaller neighborhood sizes. When null alleles are introduced, the power of detection of isolation by distance is significantly reduced and D_{CSE} remains the most powerful genetic distance. We also show that the proportion of null allelic states interact with the slope of the regression of F_{ST}/(1−F_{ST}) as a function of geographic distance. This can have important consequences on inferences that can be made from such data. Nevertheless, Chapuis and Estoup’s FreeNA correction for null alleles provides very good results in most situations. We finally use our conclusions for reanalyzing and reinterpreting some published data sets.
Introduction
The ecology of natural populations can conveniently be assessed through the study of spatiotemporal genetic variations of molecular markers with population genetics tools. This is particularly useful for organisms such as parasites and vectors that are difficult to observe directly even with mark–release–recapture methods. Indirect methods offer the possibility to access to rare and past events (Slatkin, 1987), reproductive modes and/or strategies, dispersal, population sizes and pattern of population subdivision (De Meeûs et al., 2007).
Different patterns of population subdivision can occur. The most frequent one probably corresponds to situations where geographic distance is the main factor limiting gene flow and thus allowing local genetic differentiation. This was called isolation by distance by Wright who derived the first model (Wright, 1943). Since then, different patterns of isolations by distance were proposed from the discontinuous stepping stone models, where migration mainly occurs between adjacent populations, to continuous models with different dispersal functions (see (Guillot et al., 2009) and references therein). Empirical studies of isolation by distance generally aim at determining the strength and significance of the relationship between an estimator of subdivision (for example, a genetic distance) and geographic distance between subsamples or individuals. This is typically undertaken with two distance matrices of each kind between which a correlation coefficient is computed. The significant departure from 0 of this coefficient is then tested through a randomization procedure, the Mantel test, because the existing autocorrelation that links the different cells in each matrix prevents using parametric methods (De Meeûs et al., 2007). The rows and columns of one matrix are randomized a great number of times and the Pvalue of the test is the proportion of times a correlation as big as or bigger than the one observed between the true matrices was obtained during these randomizations.
Several parameters and their estimators exist to measure genetic relationship between different subsamples. Among those, Rousset’s θ/(1−θ) (Rousset, 1997) (F_{ST}*), where θ is Weir and Cockerham’s (Weir and Cockerham, 1984) unbiased estimator of Wright’s F_{ST} (Wright, 1965) or its equivalent between individuals (Watts et al., 2007), was shown to be linearly related to geographic distances (onedimension frameworks) or to its log transform (two dimensions) with a slope that is inversely proportional to demographic parameters (effective population density and dispersal). F_{ST} is not a genetic distance in the strict sense. It is more a relative inbreeding of subsamples as compared with inbreeding in the total sample. It is thus a direct measure of inbreeding due to subdivision and hence is often considered as a measure of differentiation between subsamples. Nevertheless, θ is known to display undesirable behavior as a distance (Kalinowski, 2002) and is especially unstable (large variance) when measured between population pairs (Balloux and Goudet, 2002). These problems led to several authors advising the use of other genetic distances such as the chord distance (CavalliSforza and Edwards, 1967) (D_{CSE}), which provides more reliable relative measures for tree construction (Takezaki and Nei, 1996; Kalinowski, 2002) or the shared alleles distance (D_{SA}) (Bowcock et al., 1994), which can be preferred in some cases (De Meeûs et al., 2010) but without a formal demonstration of the statistical superiority of these distances.
Another difficulty arises when PCRbased amplification failures affect the genotyping of individuals at some loci (for example, null alleles, allelic dropouts). Null alleles lead to underestimation of genetic diversity and overestimation of population differentiation (Chapuis and Estoup, 2007) and to confusions about reproductive strategy inferences (Séré et al., 2014).
Finding methods that are the most powerful at detecting isolation by distance and the least affected by null alleles thus represents an interesting goal.
In this paper, we propose a simulation approach in order to compare the respective performances of three genetic distances, F_{ST}* (using θ and other related F_{ST} estimators), D_{CSE} and D_{SA}, in terms of statistical power for detecting isolation by distance and of robustness to the presence of null alleles. We then use the distance that displays the best performances to reanalyze real population genetics data sets and check how the initial interpretations are affected.
Methods
Simulations
Simulations were undertaken with EASYPOP version 2.0.1 (Balloux, 2001). We simulated 25, 36, 49 and 64 subpopulations of N=200 dioecious individuals with even sex ratio, random mating and discrete generations. We studied genetic information at 20 loci following a KAM model of mutation (each mutation event changes an allele into one of the K−1 remaining allelic states) with a mutation rate of u=10^{−4} and K=99. Each simulation started with all alleles present with balanced frequencies in all subpopulations (maximum diversity) and stopped after 10 000 generations (if not specified otherwise), which was considered enough for reaching mutationdrift equilibrium. Two different models of subdivided populations were explored. Island models (see De Meeûs et al., 2007), where no spatial structure is defined, provided the material for the null hypothesis (no isolation by distance). Twodimension stepping stone models (see De Meeûs et al., 2007) allowed the evaluation of the performance of the different genetic distances in case of more or less pronounced isolation by distance (alternative hypothesis), without null alleles or with increasing frequencies of those. Three migration rates (m) were used (0.01, 0.2 and 0.5). In Island models, only m=0.01 and m=0.5 were used with 100 replicates for each. For stepping stone simulations, 60 replicates were used (except if specified otherwise). For simulations with 49 and 64 subpopulations, only m=0.5 was used. For 25 subpopulations with m=0.5, for which less significant tests were expected, we undertook 80 replicates. For each replicate, 10 subpopulations with 20 individuals (10 females and 10 males) were sampled. We minimized edge effects by choosing subpopulations with four neighbors, except in the 25 subpopulations case were one subpopulation had three neighbors. Coordinates of chosen subpopulations were (row, column): (2,1) (3 neighbors), (2,2), (2,3), (2,4), (3,2), (3,3), (3,4), (4,2), (4,3), (4,4) for 25 subpopulations simulations; (2,2), (2,3), (2,4), (2,5), (3,2), (3,3), (3,4), (3,5), (4,2), (4,3) for 36 subpopulations; (2,2), (2,3), (2,4), (2,5), (2,6), (3,2), (3,3), (3,4), (3,5), (3,6) for 49 subpopulations; and (2,2), (2,3), (2,4), (2,5), (2,6), (2,7), (3,2), (3,3), (3,4), (3,5) for 64 subpopulations. A supplementary 20 replicates set was undertaken with 121 and 400 subpopulations in twodimension stepping stones to check for the effects of edges and mean number of neighbors per subpopulation on slope estimate and its variance. With n subpopulations, the mean number of neighbors for each subpopulation is . With 400 subpopulations, the mean number of neighbors (3.8) is then reasonably close to the one in a torus (4) and hence the effect of marginal subpopulations becomes more discrete. For n=400, two sample sets were analyzed: central set with subpopulations at coordinates (9,8), (9,9), (9,10), (9,11), (9,12), (10,8), (10,9), (10,10) (10,11) and (10,12); and Marginal set with populations at coordinates (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9) and (1,10). For these 20 replicates, migration rate was m=0.5 and the number of generation only 200, which was enough for reaching equilibrium for F_{ST}. Other parameters were the same as for other simulations. Graphic representations of the different sampling designs can be found in Supplementary File S1. The effect of reducing the number of possible alleles was also explored with K=2, u=10^{−9} (to conform to singlenucleotide polymorphisms) (Séré et al., 2014), n=25 subpopulations and m=0.5. In that case, we used 20 loci to compare with other cases and 200 loci to conform to what could be advised for singlenucleotide polymorphisms. We also undertook some simulations with K=5 and u=10^{−7}, K=10 and u=10^{−6}, K=20 and u=10^{−5} and K=40 and u=10^{−4}. This was carried out for m=0.5 and n=36 or n=25. Finally, variable subpopulation sizes were explored with subpopulation sizes alternating between 100 and 200 individuals in the 36 subpopulation case (the most discriminating between genetic distances, see below) with the same other parameters as described above.
Null alleles
We define null alleles as alleles that cannot be seen (as it occurs through PCR amplification failure). At a given locus, an individual can thus be homozygous or heterozygous for normal alleles, homozygous for the same null allele or heterozygous for two different null alleles, in which cases the individual corresponds to a missing data (blank) at the corresponding locus. It can be heterozygous for a normal and a null allele, in which case the individual is seen as a homozygous for the normal allele. In simulations, null alleles were generated by recoding the data outputted by EASYPOP. 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 two 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 produces a large variation of null allele frequencies across loci (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). These proportions of null alleles (10, 20 and 50%) actually correspond to the proportions of allelic states that are null. For the sake of simplification, we will call these ‘Proportion of null alleles’ or ‘Null allele frequencies’.
Geographic and genetic distances
Geographic distances (D_{G}) between each subpopulation pairs were computed with the Euclidian distance using their coordinates as implemented in Genepop version 4.6 kindly provided by F Rousset before online publication (updated from Rousset, 2008).
We have used three genetic distances among the most popular ones: D_{CSE} (CavalliSforza and Edwards, 1967), D_{SA} (Bowcock et al., 1994), and F_{ST}*=F_{ST}/(1−F_{ST}) (Rousset, 1997). Other F_{ST}related measures were also studied in a subset of simulated data sets: G_{ST}′ (Meirmans and Hedrick, 2011) and D_{J} (Jost, 2008), computed with GenAlEx 6.5 (Peakall and Smouse, 2012); Robertson and Hill’s F_{ST_RH} and its modified version F_{ST_RH}′ (Raufaste and Bonhomme, 2000), computed with Genetix (Belkhir et al., 2004). D_{CSE} and D_{SA} were computed with MSA v3.1 (Dieringer and Schlötterer, 2003). In one instance, MSA could not compute one genetic distance for unknown reasons. This distance was recomputed with FreeNA (Chapuis and Estoup, 2007). We then used Genepop to compute F_{ST}*=θ/(1−θ), where θ is Weir and Cockerham’s unbiased estimator of F_{ST} (Weir and Cockerham, 1984). In twodimension isolation by distance, the slope b of the regression F_{ST}*=a+b × ln(D_{G}) is inversely proportional to the average neighborhood size of a subpopulation: b=1/(4πD_{e}σ^{2}), where D_{e} is the effective subpopulation density and σ^{2} is the average squared axial distance between an adult and its parents (dispersal) (Rousset, 1997) and hence σ is half the average adult–parent distance.
Detecting isolation by distance
For each data set, not recoded or recoded for null allele presence, correlation between geographic and genetic distances was computed and tested with a Mantel test with Genepop and 1 000 000 randomizations, as recommended (De Meeûs et al., 2007). The test uses Pearson’s regression coefficient between geographic and genetic distance matrices as the statistics and the Pvalue of the test corresponds to the proportion of times the randomized regression coefficient is as big as or bigger than the observed one. For the sake of homogeneity, all Mantel tests were undertaken with the ln transform of geographic distances. Even if the Pvalues obtained with the untransformed distances can be different, this did not affect the proportion of significant tests obtained (not shown). The proportion of significant tests at the 5% level was used to compare performances between the different genetic distances and between data sets with or without null alleles. Genepop also performs an ABC Bootstrap procedure that allows computing a 95% confidence interval (CI) for the regression slope. For each data set, in addition to Mantel tests using genetic distances, we have also recorded when this CI excluded the 0 value as a significant test at the 5% level (95% CI).
For simulations with the Island model (no spatial structure), to check if no >5% significant tests were obtained with each genetic distance, we undertook a unilateral exact binomial test with R version 3.2.1 (R Development Core Team, 2016).
Performance comparisons between genetic distances were carried out with Fisher’s exact tests with R. To account for multiple nonindependents testing, the Pvalues obtained were adjusted with the sequential Bonferroni procedure (for example, De Meeûs, 2014).
The effect of null alleles on the power to detect isolation by distance was assessed with generalized mixed models with a binomial error distribution and a logit link with the package lme4 of R (Bates et al., 2015). The model (command glmer) had the form:
where, for ‘Significance’, a value of 0 was given if P>0.05 and a value of 1 was given if P⩽0.05, ‘GenetDist’ is a factor defining one of the three genetic distances, ‘PropNulls’ is the proportion of null alleles as defined above, ‘m’ is the migration rate, ‘(1Rep)’ are the random effects of replicates and ‘:’ stands for interaction between two variables.
To check whether collinearity between explanatory variables might explain convergence failure for the model above, significance cooccurrence between statistics (for example, between F_{ST}* and D_{CSE}) was measured by recoding the results as 1 (not significant) and 2 (significant) for each statistic and doubling these coding (11 or 22) to build an artificial genotypic data set with four ‘loci’ (the four statistics) with two alleles each (hence if F_{ST}* and D_{CSE} gave a significant isolation by distance this would result in genotypes 22 and 22). We then computed Garnier Géré and Dillman correlation coefficient r_{GGD} (GarnierGéré and Dillmann, 1992) between each ‘locus’ pair. This correlation was computed in Genetix 4.05.2 (Belkhir et al., 2004).
Effects of null alleles on the regression slope of isolation by distance
The effect of null alleles and migration rate on the slope b, of the regression F_{ST}*=a+b × ln(D_{G}), was assessed with a linear mixed effect model (lme) (Pinheiro et al., 2014), fitted with the maximum likelihood method, with package nlme of R version 3.0.2 and a model of the type:
where the random effects are the replicates (random=~1Rep). The variance in each replicate increases as a function of the proportion on null alleles and migration rates. To account for this violation of homoscedasticity, we used the argument ‘weights’ (weights=varIdent(form=~1PropNulls) or weights=varIdent(form=~1m)), as recommended in the R documentation (Pinheiro and Bates, 2000). Because the slope (bounded between −1 and +1) cannot strictly follow a normal distribution, we also undertook Spearman correlation tests with the package rcmdr (Fox, 2005) of R for each migration rate separately.
The effect of the proportion of null alleles and migration rate on the variance of the slope of the aforementioned regression was analyzed with a linear regression with R and the model:
The quality of fit of each of aforementioned model was assessed by graphically checking residuals’ normality and homoscedasticity. For each model, the minimal model was looked for with the command ‘drop1’, using the Akaike information criterion (AIC; Akaike, 1974).
Comparative vulnerability of the different genetic distances to null alleles
For each genetic distance, the coefficient of determination (R^{2}, the proportion of variance explained by the model) of the regressions between ln(D_{G}) matrix and genetic distances matrices with null alleles of increasing frequencies (0, 0.1, 0.2 and 0.5) was computed for each replicate of each parameter set. We used rcmdr to compute R^{2}. For each parameter set, in terms of null allele frequency and migration rate, and for each genetic distance, we computed the average R^{2} () over replicates and used the s.d. of the mean and Student’s t_{0.05,γ}, where γ=Nrep−1 (Nrep=number of replicates of a given simulation), to compute the 95% CIs of each as . We then applied a generalized mixed model to the data, as described above with a random effect of replicates (random=~1Rep), fitted with the maximum likelihood method (method=‘ML’), adjusted for heteroscedasticity with the argument ‘weights=varIdent(form=~1PropNulls)’ or ‘weights=varIdent(form=~1m)’ (model with the lowest AIC was kept). The model explored was:
Here again the function drop1 was used to try selecting the best (minimal) model.
Using corrected estimates
We also checked how far results can be improved using the available correcting option implemented in the software FreeNA (Chapuis and Estoup, 2007). Genetic parameters are estimated from the corrected data set (replacing all missing genotypes by homozygous genotypes for the same null allele labeled 999) only on visible alleles and using the ENA correction method for F_{ST} and using the INA correction method for D_{CSE} (Chapuis and Estoup, 2007). We also used these estimations on several simulated data sets with null alleles to check for the effect on the slope and on power of isolation by distance testing with regression approaches.
Real data
Several available data sets using microsatellite markers were reanalyzed to illustrate our results. For all data, we transformed GPS coordinates (Genepop uses coordinates of subsamples or individuals) into UTM so that we get geographic distances in meters. The first data set was on the European vector of Lyme disease, the tick Ixodes ricinus (De Meeûs et al., 2002) where a lot of null alleles are present (frequently 40–50% and up to 60 %). The second concerned the rusa deer Cervus timorensis russa in NewCaledonia (De GarineWichatitsky et al., 2009; without null allelles). The third and fourth data came from (Kone et al., 2011) on two tsetse flies (trypanosomosis vectors), Glossina tachinoides and G. palpalis gambiensis in Burkina Faso, where as many as 40–50 % of null alleles can be found and up to 70%. The fifth data concerned another tsetse fly, G. palpalis from Cameroon and from the Democratic Republic of Congo (DRC), (Mélachio et al., 2011) where null allele proportions frequently reach 40–50% and up to 70%. The sixth data set concerned Leishmania guyanensis (a trypanosomatid parasites) from French Guyana (Rougeron et al., 2011) without null alleles. For I. ricinus and tsetse fly data, because some loci are Xlinked, we only used females so that FreeNA results are not too biased by male genotypes at these loci. These data sets were all analyzed for isolation by distance using Genepop using F_{ST}* and D_{CSE} on raw data and on FreeNAcorrected data. We then compared the Pvalues obtained and also the neighborhood size as Nb=1/b (b is the slope of the isolation by distance regression) (Rousset, 1997). For L. guyanensis, data could only be analyzed at an individualbased scale. We used the statistic a (Watts et al., 2007) as recommended when the slope b obtained is such that neighborhood Nb=1/b<50. FreeNA cannot compute this statistic so that the effect on the slope could not be assessed here. Three independent samples (2006, 2007 and 2008) were analyzed separately and then the resulting Pvalues combined with generalized binomial procedure with MultiTest V 1.2 (De Meeûs et al., 2009) setting the threshold number of tests k′=3 as recommended when the total number of tests is <4 (De Meeûs, 2014). Some data sets did not have strong evidence for null alleles (rusa deer and Leishmania) while other displayed strong evidence for null alleles (ticks, tsetse flies) with other problems, such as probable Wahlund effects (ticks and G. palpalis gambiensis) or short allele dominance (ticks). Leishmania were shown to alternate clonal propagation with sexual recombination with a strong tendency for mating between relatives (inbreeding).
Results
For all simulations with 10 000 generations (25–64 subpopulations), the final mean number alleles maintained and unbiased estimate of genetic diversity H_{S} (Nei and Chesser, 1983) varied between 17 and 37 and between 0.58 and 0.84, respectively (see Supplementary File S1), which should provide unbiased estimates of D_{e}σ^{2} (Leblois et al., 2003). For bigger simulations (121 and 400 subpopulations) that were run for only 1000 and 200 generations only, values obtained are far beyond the limit and are expected to give biased estimates. In fact, only the case with 121 subpopulations provided biased slope estimates (Supplementary File S1, worksheet ‘Compare slopes’).
Detecting isolation by distance under the null hypothesis
In the Island model, where no spatial structure exists, no procedure gave >5% significant isolation by distance signature (all Pvalues >0.2, exact unilateral binomial tests). With 11 (5.5 %), 7 (3.5 %), 13 (6.5 %) and 8 (4 %) significant tests (out of 200 tests) for 95% CI for the regression slope, F_{ST}*, D_{CSE} and D_{SA} respectively, no procedure gave more often significant tests than the other (Pvalue=0.5047, Fisher exact test; Supplementary File S1). This confirms the robustness of Mantel test. With between 3.5% and 6.5 % significant tests under the null hypothesis, it also highlights that with such data sets Mantel test is not too conservative.
Detecting isolation by distance under the alternative hypothesis
For the stepping stone model, results are presented in Figure 1. Detailed simulation results are available in Supplementary File S1. The best results are observed for D_{CSE}. Globally, summing all significant values across the 380 simulations for all genetic distances, all comparisons are significant after sequential Bonferroni correction. The poorest statistic is the test based on 95% CI of the slope (95% CI) (187 significant tests), followed by F_{ST}* (217 significant tests), D_{SA} (257 significant tests) and D_{CSE} (293 significant tests). Except when migration rates are low, where the different statistics tend to perform equally well, difference in performance between the different genetic distances follows the same pattern independently of the number of subpopulations: 95% CI<F_{ST}*<D_{SA}<D_{CSE}. We checked this in the most discriminating simulations in stepping stone models of 36 subpopulations. In Supplementary File S1 (worksheet ‘OtherDistances’), it can be seen that the Pvalues obtained for isolation by distance with G_{ST}′ and D_{J} are almost completely correlated to those obtained with F_{ST}* (r>0.889) and thus offer no benefit for isolation by distance detection. Alternatively, F_{ST_RH} appears more performant than all other statistics but D_{CSE}, while F_{ST_RH}′ appears just above F_{ST}*. From the Supplementary File S1, it can be seen that the regression slope (b) obtained with F_{ST_RH} is negatively biased as compared with Weir and Cockerham’s based F_{ST}* and that this bias increases with F_{ST}, while its variance is much smaller one, as expected (Raufaste and Bonhomme, 2000). It can also be seen that F_{ST_RH}′ displays a constant positive bias and the highest variance, which explains its poor performances.
Effect of subpopulation number on the slope and its variance
There is a negative, though marginally not significant (Pvalue=0.0599) relationship between the slope and the number of subpopulations, with a decrease of the positive bias of b (from ~0.003 for n=25 to ~0.002 for n=64) as compared with the expected value b_{e}=1/2πNm (Rousset, 1997) (0.00159 when N=200 and m=0.5) with decreasing subpopulation number. This may come from the mean number of neighbors per subpopulation, which tends to increase with the total number of subpopulations, as the proportion of marginal subpopulations decreases as well. To confirm this, we have studied the slope and its variance in the center and the margin of 20 × 20=400 subpopulations of 200 individuals in a stepping stone with m=0.5 (10 replicates). The mean slope observed for central subpopulations perfectly fits the expected value and is significantly above for marginal subpopulations (Supplementary File S1). At the same time, we can observe a swift decrease of 95% CI with the number of subpopulations. As a consequence, the proportion of significant tests tends to increase with the number of subpopulations n. Nevertheless, the relative performance of each statistic, as compared with the others, remains unchanged and hence D_{CSE} stays the best statistic for detecting isolation by distance.
Reduced number of possible alleles on statistical power
When the number of available alleles is K=2, there is no more significant differences in performances between the different statistics, which all show weak power to detect isolation by distance (only 11% tests are significant with D_{CSE} against 50% when K=99). At the same time, averaged slopes are not affected but the variance doubled (Supplementary File S1). With the more realistic 200 loci, the power to detect isolation by distance and lower variance for slope are restored. Nevertheless, no genetic distance seems better than the other in that configuration (Supplementary File S1).
Effect of variable subpopulation size on the power of tests
In that case, D_{CSE} still displays the highest power but is not significantly better than D_{SA} anymore in that respect (Supplementary File S1).
Effect of null alleles on the power to detect a signature of isolation by distance
We have already seen that the number of subpopulations does not influence statistics’ relative performances in detecting isolation by distance. Nevertheless, small subpopulation numbers increases the variance of these statistics and hence decreases the power to detect isolation by distance for all them. Consequently, the effect of null alleles was only studied with n=25 stepping stone models. The best model and significance of variables of the generalized linear mixed model are presented in Supplementary Table S1. The model could never converge when 95% CI was included, as a probable result of its collinearity with other statistics as can be measured with Garnier Géré and Dillman r_{GGD} (see Supplementary File S1). We also needed to remove two interaction terms (m:PropNulls and m:GenetDist). After model comparisons with the AIC, the minimum model retained is Significance~PropNulls+m+GenetDist+(1Rep). The results obtained with the generalized mixed effect model show that F_{ST}* has far less power to detect isolation by distance than D_{CSE}, and D_{SA} has slightly less power than D_{CSE} (Supplementary Table S1 and Supplementary File S2). As expected, migration rate has a major negative effect. The power of detection of isolation by distance signatures is strongly negatively correlated with the proportion of null alleles. The interaction between genetic distances and the proportion of null alleles was not important for the model (lower AIC without it) and was thus removed from the model. Hence, the performance differences between all distances do not significantly change with null alleles.
Effect of null alleles on the slope of Rousset’s regression
The analysis of this model is presented in Supplementary Table S2 and in Supplementary File S2. Controlling for the increasing variance of the residuals with null allele frequencies or migration rates (weights=varIdent(form=~1PropNulls) or weights=varIdent(form=~1m)) provided identical results for the linear mixed effect model, though AIC was smaller with the second model. All parameters and interactions appeared highly significant (Supplementary Table S2). The corresponding results of Spearman correlation (ρ) test results are given in Figure 2. It can be seen that there is a significant positive effect of null allele frequencies on the slope (b) of the regression F_{ST}*= a+b × ln(D_{G}) for the smallest neighborhoods (m⩽0.2) and a reverse relationship for the biggest neighborhood (m=0.5) (interaction between m and null allele frequency). The three Spearman’s rank correlation tests (for each migration rate explored) were all highly significant.
There is also an increase of the variance with null allele frequencies when neighborhoods are small but not for the biggest ones (m=0.5) (Supplementary Table S3 and Figure 2) (see corresponding graphics in Supplementary File S2). The significant interaction between m and the proportion of null alleles makes the effect of m very uneasy to predict and is not significant with the ttest. An analysis of variance does not confirm the significance of null alleles proportion (m becomes significant). Thus the interaction at least is significant (Supplementary Table S3).
Vulnerability of the different genetic distances to null alleles
Results of the fixed effect model are given in Supplementary Table S4. The smallest AIC was obtained for the model with weights=varIdent(form=~1m). Dropping terms lead to remove the interaction between null allele frequencies and genetic distances and all other parameters were highly significant (Supplementary Table S4). The proportion of null alleles and migration rates are the most important factors and D_{CSE} is associated with the highest R^{2}, followed by D_{SA} and then F_{ST}*. All parameters significantly interact with m: the difference between genetic distances is the highest for low migration rates and low null allele frequencies and the magnitude of differences tends to drop for highest values (see Figure 3 and Supplementary File S2).
Using FreeNa corrections
Simulations were revisited with the correction implemented in FreeNA. In the Figure 4 (see also Supplementary File S2), it can be seen that FreeNA provides a very good correction for the slope, except when the proportion of null alleles is too big (50%) and isolation by distance weak (m=0.5), because the variance of estimate becomes very large, thus reducing R^{2} to a great extent with a slight alteration of slope estimate (Figure 4). A logistic mixed model on the significance obtained with FreeNAcorrected data was undertaken using D_{CSE}. It suggests that using FreeNAcorrected data does not bring more power and even less so (Supplementary Table S5). There is also a small effect of interactions of statistics with the other variables, suggesting that for some migration rates and/or null allele frequencies FreeNAcorrected data may be slightly more powerful, but the effect is obviously weak as compared with the effect of null alleles alone (see also Supplementary File S3).
Real data
Results are presented in Supplementary File S3. Generally speaking, as for the simulation results, FreeNA does not bring much benefit for the significance of the tests and D_{CSE} from unmodified data provided more power to the tests, except for G. palpalis palpalis from Cameroon and DRC and G. palpalis gambiensis from BurkinaFaso.
For I. ricinus females from Switzerland, even if the drop in Pvalue is substantial (from 0.241 to 0.088), the use of D_{CSE} leads to a marginally not significant isolation by distance. The use of FreeNA provides a strong effect as it leads to almost halve Nb (from 707 to 444 individuals) (hence in the wrong direction).
For Cervus timorensis russa in NewCaledonia, using D_{CSE} changes the significance level from 0.15 to 0.008 and, as expected with a data set with no or almost no null alleles, neighborhood size is weakly affected when using FreeNA correction (from 104 to 107).
For Glossina tachinoides females in Burkina Faso, the use of D_{CSE} is important (the Pvalue drops from 0.1 to 0.005) and FreeNA has much effect on Nb that increases from 81 to 1667.
For G. palpalis palpalis from Cameroon and DRC, F_{ST}*based tests provide highly significant results while D_{CSE}based tests give marginally not significant Pvalues (0 and 0.146, respectively). The FreeNAcorrected Nb is weakly increased (from 153 to 179).
For Glossina palpalis gambiensis females from Burkina Faso, the smallest Pvalue is obtained with F_{ST}* on FreeNAadjusted data (0.004) while D_{CSE} does not give a significant result (0.522). Here Nb estimated from FreeNAcorrected data decreases from 249 to 196, instead of increasing.
Finally, in L. guyanensis data from French Guyana, the Pvalue obtained with a (3 years combined) drops from a slightly significant value with F_{ST}* (0.017) to a highly significant value when using D_{CSE} (0.0018).
Discussion
Our Island model simulations have confirmed the robustness of Mantel tests as no >5% significant tests were found in that case.
In twodimension stepping stone models, the most powerful statistic to detect isolation by distance is D_{CSE}, followed in order of power by F_{ST_RH}, D_{SA}, F_{ST_RH}′, F_{ST}* (together with G_{ST}′ and D_{J}) and the least powerful being the test based on the 95% CI of the slope. The fact that F_{ST_RH} provides more powerful tests than the other statistics but D_{CSE} is probably linked to its very low variance for small values (Raufaste and Bonhomme, 2000), which concerns most of our simulations. The greater weight given to the rarest alleles is probably the main cause of this, though unbalanced samplings, which were not fully explored here, might lower the good properties of this statistic in this respect (Goudet et al., 1996). Similar reasons probably explain the greater power of D_{CSE} (Kalinowski, 2002), though its connection with F_{ST}based measures is more complex.
When subdivision is extreme (m=0.01), choice of a genetic distance to detect a significant signature of isolation by distance becomes irrelevant (all statistics behave equally well). The same lack of difference occurs for weakly polymorphic loci as when K=2 where at least 200 loci are required for the sake of statistical power and slope estimate accuracy. This is more than classically advised given our sample sizes (20) (for example, Smouse, 2010). In fact, it can be seen from Supplementary File S5 that D_{CSE} outperforms the other statistics only when genetic diversity outreaches 0.4–0.5 (hence applying to microsatellite markers). The difference of statistical power may thus be related to the way allele frequencies are weighted by each statistic. Nevertheless, it is somehow difficult to understand this when more than two alleles are maintained. It will thus require further theoretical investigations to be fully understood. Performance ranking is not influenced by the number of subpopulations, subpopulation size variation or by the proportion of null alleles. The poor performances of F_{ST}based measures is a confirmation of earlier studies where it was shown that its weak statistical performances results from an inflated variance of its unbiased estimator θ especially when measured between paired subsamples (Balloux and Goudet, 2002). Intrinsically, F_{ST} is more a measure of inbreeding within subsamples than a genetic distance between subsample pairs.
Another interesting and unexpected result is the influence of marginal subpopulations, located at the edges of the metapopulation. Because these marginal subpopulations receive immigrants from less numerous neighbors, this results in an increase of estimated slope when these marginal subpopulations are included in the sample or at the vicinity of those. Moreover, because in small metapopulations central subpopulations are more closely related to these marginal subpopulations, this slope inflation will also affect the entire mesh. This also increases the heterogeneity between different subpopulations and thus the variance of slope estimates and hence the power of isolation by distance tests. Consequently, sampling at the margins of any metapopulation will result in a reduced neighborhood size as compared with central subpopulations, and sampling small metapopulation (25–64 subpopulations) will also result in reduced effective neighborhood size and decline the power of isolation by distance detection (because of increased variance).
The presence of null alleles disturbs the signature of isolation by distance not only in terms of detection but also in terms of the slope of the regression that can be used for demographic inferences. The decrease of power detection affects each distance equally. More importantly, null alleles generate an inflated estimate of the slope and of its variance. With regard to demographic inferences, null alleles will therefore lead to an underestimate of neighborhood sizes and/or of dispersal but with an inflated uncertainty as well.
Chord distance D_{CSE} is known to better reflect the topology of branching nodes between subsamples or species (Takezaki and Nei, 1996). We confirm here that D_{CSE} also reflects geographic distances more accurately than D_{SA} and F_{ST} and is also the most robust genetic distance to null allele interference in that respect.
Null alleles are known to have major effects when differentiation is strong and much weaker effects when it is weak (Chapuis and Estoup, 2007). We confirm this in the isolation by distance case, with an additional interactive effect. Indeed, in case of high migration rate (m=0.5), weak null allele effect on F_{ST}* seems to interact strongly with its variance resulting in a decrease of the regression slope of isolation by distance with null allele frequencies, while it increases with those in case of smaller neighborhoods (for example, m=0.2).
FreeNA correction (Chapuis and Estoup, 2007) appears quite satisfactory if not excellent, at least for slope estimates, except for high null allele frequencies and strong migration rates where the variance of estimates might provide unreliable results. Statistically speaking though, this correction brings weak improvements if any. Nevertheless, the underlying hypothesis of FreeNA is that observed heterozygote deficits must be entirely explained by null alleles. Other factors can produce heterozygote deficits as measured by F_{IS}. Wahlund effects and selfing should affect all loci equally in this respect and consequently should theoretically be easy to distinguish from null alleles that affect each locus in a particular way. Nevertheless, null alleles are not specific to panmictic populations and can as well affect selfing species or subsamples experiencing Wahlund effects. Some methods exist for quantifying the relative effects of inbreeding and null alleles (for example, see Campagne et al., 2012 and references therein). In a simulation with the same parameters as described above, with 20% of selfing and m=0.01, the use of FreeNA produced an underestimate of the isolation by distance slope of 6% in average, which is not much and thus validates the usefulness of FreeNA in that case. Wahlund effect might be more detrimental and will require further attention. Indeed, Wahlund effect will decrease identity between individuals in subsamples and can also increase identity between subsamples resulting in an artificial deflation of genetic differentiation between geographically separated groups. Hence, Wahlund effect is likely to decrease the slope of isolation by distance regressions (and hence increase neighborhood estimate). Applying FreeNA might induce an additional decrease of this slope and lead to biased results. Important Wahlund effects are known to occur in I. ricinus tick populations in Europe (Kempf et al., 2010). This might represent a good explanation why D_{CSE} did not produce a significant isolation by ditance. Nevertheless, the fact that FreeNA correction had such a spectacular decreasing effect on the neighborhood estimate in that particular situation remains obscure. Short allele dominance affecting one of the five loci and high null allele frequencies in four loci probably explains these discrepancies. Wahlund effects are also known to occur in G. palpalis gambiensis and might explain the observed lack of improvement when using D_{CSE}. The results obtained for the rusa deer and Leishmania are in total agreement with what was expected: an increased power of isolation by distance test using D_{CSE}, changing the initial rusa deer Pvalue into a significant one, and for this last, lacking null alleles, an absence of effect of FreeNA correction. For G. palpalis palpalis, for which Wahlund effect has been excluded as a possible explanation, the reason why the reverse result was obtained for significance ranking between F_{ST}* and D_{CSE} might be just due to chance. In our simulation, F_{ST}*based tests could rarely, but possibly, give much smaller Pvalues than D_{CSE}based ones. Finally, for G. tachinoides, performances of the different statistics are in the expected direction as is FreeNAcorrected neighborhood size estimate, though the decrease is a bit extreme. Wahlund effect was not detected in that species but may be this would require further investigations.
Finally, onedimension patterns of migration, which are known to structure more than twodimension ones (Rousset, 1997; Séré et al., 2014), were not studied here but would probably have provided more frequent isolation by distance detection as a whole. Nevertheless, we see no reasons why D_{CSE} would not remain above the others in that respect, even if this might require further investigations. More complex migration models could also have been explored. This would have required the choice of supplementary parameters (distance of dispersal and kurtosis) with little gain in generalization of the results as we see no obvious reason why this would dramatically change the major conclusions of the present study.
Recommandations
With null allelefree data, using D_{CSE} chord distance (CavalliSforza and Edwards, 1967) represent the best strategy in most situations for detecting isolation by distance, except for low polymorphism situations (H_{S}<0.4) where the choice of any statistics seems irrelevant.
To cite a verbal quotation from a lecture: ‘the best way to handle missing data is to have none’ (literally translated from a lecture given by JeanDominique Lebreton). However, sometimes the worse is inevitable. In case of null alleles, if these are not too frequent (no >20%) and if one can be confident that null alleles represent the only explanation for observed heterozygote deficits, FreeNA correction will provide an excellent way to correct for slope biases. As null alleles equally affect the statistical power of all genetic distances, D_{CSE} stays the best option in that case.
Finally, when other factors than null alleles might be responsible for observed heterozygote deficits, available amending method will not be always useful. In such situations, one should consider that null alleles, if evidenced from important variance of Wright’s F_{IS} across loci, will produce a reduced probability to find a significant isolation by distance. In that case, resampling at smaller scales (to avoid or limit Wahlund effects) and redesigning primers (to reduce the consequences of null alleles) might be of help.
Data archiving
All simulation results and real data can be found in Supplementary Files S1–S5. All simulated data can be sent upon request to the corresponding author.
References
Akaike H . (1974). A new look at the statistical model identification. IEEE Trans Auto Control 19: 716–723.
Balloux F . (2001). EASYPOP (version 1.7): a computer program for population genetics simulations. J Hered 92: 301–302.
Balloux F, Goudet J . (2002). Statistical properties of population differentiation estimators under stepwise mutation in a finite island model. Mol Ecol 11: 771–783.
Bates D, Maechler M, Bolker B, Walker S . (2015). Fitting linear mixedeffects models using lme4. J Stat Softw 67: 1–48.
Belkhir K, Borsa P, Chikhi L, Raufaste N, Bonhomme F . (2004) GENETIX 4.05, Logiciel Sous Windows TM Pour la Génétique des Populations. Laboratoire Génome, Populations, Interactions, CNRS UMR 5000. Université de Montpellier II: Montpellier, France.
Bowcock AM, RuizLinares A, Tomfohrde J, Minch E, Kidd JR, CavalliSforza LL . (1994). Highresolution of human evolutionary trees with polymorphic microsatellites. Nature 368: 455–457.
Campagne P, Smouse PE, Varouchas G, Silvain JF, Leru B . (2012). Comparing the van Oosterhout and ChybickiBurczyk methods of estimating null allele frequencies for inbred populations. Mol Ecol Res 12: 975–982.
CavalliSforza LL, Edwards AWF . (1967). Phylogenetic analysis: model and estimation procedures. Am J Hum Genet 19: 233–257.
Chapuis MP, Estoup A . (2007). Microsatellite null alleles and estimation of population differentiation. Mol Biol Evol 24: 621–631.
De GarineWichatitsky M, De Meeûs T, Chevillon C, Berthier D, Barre N, Thévenon S et al. (2009). Population genetic structure of wild and farmed rusa deer (Cervus timorensis russa in NewCaledonia inferred from polymorphic microsatellite loci. Genetica 137: 313–323.
De Meeûs T . (2014). Statistical decision from k test series with particular focus on population genetics tools: a DIY notice. Infect Genet Evol 22: 91–93.
De Meeûs T, Beati L, Delaye C, Aeschlimann A, Renaud F . (2002). Sexbiased genetic structure in the vector of Lyme disease, Ixodes ricinus. Evolution 56: 1802–1807.
De Meeûs T, Guégan JF, Teriokhin AT . (2009). MultiTest V.1.2, a program to binomially combine independent tests and performance comparison with other related methods on proportional data. BMC Bioinformatics 10: 443.
De Meeûs T, Koffi BB, Barré N, de GarineWichatitsky M, Chevillon C . (2010). Swift sympatric adaptation of a species of cattle tick to a new deer host in NewCaledonia. Infect Genet Evol 10: 976–983.
De Meeûs T, McCoy KD, Prugnolle F, Chevillon C, Durand P, HurtrezBoussès S et al. (2007). Population genetics and molecular epidemiology or how to "débusquer la bête". Infect Genet Evol 7: 308–332.
Dieringer D, Schlötterer C . (2003). Microsatellite analyser (MSA): a platform independent analysis tool for large microsatellite data sets. Mol Ecol Notes 3: 167–169.
Fox J . (2005). The R commander: a basic statistics graphical user interface to R. J Stat Softw 14: 1–42.
GarnierGéré P, Dillmann C . (1992). A computer program for testing pairwise linkage disequilibria in subdivided populations. J Hered 83: 239.
Goudet J, Raymond M, De Meeûs T, Rousset F . (1996). Testing differentiation in diploid populations. Genetics 144: 1933–1940.
Guillot G, Leblois R, Coulon A, Frantz AC . (2009). Statistical methods in spatial genetics. Mol Ecol 18: 4734–4756.
Jost L . (2008). GST and its relatives do not measure differentiation. Mol Ecol 17: 4015–4026.
Kalinowski ST . (2002). Evolutionary and statistical properties of three genetic distances. Mol Ecol 11: 1263–1273.
Kempf F, McCoy KD, De Meeûs T . (2010). Wahlund effects and sexbiased dispersal in Ixodes ricinus, the European vector of Lyme borreliosis: new tools for old data. Infect Genet Evol 10: 989–997.
Kone N, Bouyer J, Ravel S, Vreysen MJB, Domagni KT, Causse S et al. (2011). Contrasting population structures of two vectors of African trypanosomoses in Burkina Faso: consequences for control. PLoS Negl Trop Dis 5: e1217.
Leblois R, Estoup A, Rousset F . (2003). Influence of mutational and sampling factors on the estimation of demographic parameters in a "Continuous" population under isolation by distance. Mol Biol Evol 20: 491–502.
Meirmans PG, Hedrick PW . (2011). Assessing population structure: FST and related measures. Mol Ecol Res 11: 5–18.
Mélachio T, Tito, Tanekou, Simo G, Ravel S, De Meeûs T et al. (2011). Population genetics of Glossina palpalis palpalis from central African sleeping sickness foci. Parasit Vectors 4: 140.
Nei M, Chesser RK . (1983). Estimation of fixation indices and gene diversities. Ann Hum Genet 47 (Pt 3): 253–259.
Peakall R, Smouse PE . (2012). GenAlEx 6.5: genetic analysis in Excel. Population genetic software for teaching and research  an update. Bioinformatics 28: 2537–2539.
Pinheiro JC, Bates DM . (2000) MixedEffects Models in S and SPLUS. SpringerVerlag: New York, USA.
Pinheiro JC, Bates DM, DebRoy S, Sarkar D, R Core Team. (2014). nlme: linear and nonlinear mixed effects models. R package version 31117 ed 49.
R Development Core Team. (2016). R Foundation for Statistical Computing: Vienna, Austria Available at: http://www.Rproject.org.
Raufaste N, Bonhomme F . (2000). Properties of bias and variance of two multiallelic estimators of FST . Theor Popul Biol 57: 285–296.
Rougeron V, Bañuls AL, Carme B, Simon S, Couppie P, Nacher M et al. (2011). Reproductive strategies and population structure in Leishmania: substantial amount of sex in Leishmania Viannia guyanensis. Mol Ecol 20: 3116–3127.
Rousset F . (1997). Genetic differentiation and estimation of gene flow from Fstatistics under isolation by distance. Genetics 145: 1219–1228.
Rousset F . (2008). GENEPOP’007: a complete reimplementation of the GENEPOP software for Windows and Linux. Mol Ecol Res 8: 103–106.
Séré M, Kaboré J, Jamonneau V, Belem AMG, Ayala FJ, De Meeûs T . (2014). Null allele, allelic dropouts or rare sex detection in clonal organisms: simulations and application to real data sets of pathogenic microbes. Parasit Vectors 7: 331.
Slatkin M . (1987). Gene flow and the geographic structure of natural populations. Science 236: 787–792.
Smouse PE . (2010). How many SNPs are enough? Mol Ecol 19: 1265–1266.
Takezaki N, Nei M . (1996). Genetic distances and reconstruction of phylogenetic trees from microsatellite DNA. Genetics 144: 389–399.
Watts PC, Rousset F, Saccheri IJ, Leblois R, Kemp SJ, Thompson DJ . (2007). Compatible genetic and ecological estimates of dispersal rates in insect (Coenagrion mercuriale: Odonata: Zygoptera) populations: analysis of ‘neighbourhood size’ using a more precise estimator. Mol Ecol 16: 737–751.
Weir BS, Cockerham CC . (1984). Estimating Fstatistics for the analysis of population structure. Evolution 38: 1358–1370.
Wright S . (1943). Isolation by distance. Genetics 28: 114–138.
Wright S . (1965). The interpretation of population structure by Fstatistics with special regard to system of mating. Evolution 19: 395–420.
Acknowledgements
This study was financed by the AIRD (Agence Interétablissements de Recherche pour le Développement) as part of a PEERS program (Programme d’Excellence pour l’Enseignement et la Recherche au Sud) TAO (Trypanosomes et tsétsés en Afrique de l’Ouest: apport de la génétique des populations). It was undertaken in the Lamivect (Laboratoire Mixte International sur les maladies à vecteurs de BoboDioulasso), which is supported by the IRD (Institut Recherche pour le Développement). MS is financed by a grant from the AIRD ‘thèse du programme d’appui au renforcement et à la consolidation des potentiels de recherche dans les pays du Sud’ with the cooperation of the Centre de coopération Internationale en Recherche Agronomique pour le Développement (CIRAD). We thank Dr Sophie Ravel, Dr Bruno Bucheton, Dr Christine Chevillon and Dr Frédéric Simard for very useful discussions during PhD committees. A special thank goes to François Rousset and his patience while discussing many different issues regarding some odd results and for kindly providing Genepop version 4.6 before online publication. We also thank Giuliano Cecchi and Jérémy Bouyer for their help during UTM transformation of GPS coordinates from two different UTM zones. We finally thank Heredity’s Editor and four anonymous referees whose comments helped to considerably improve our manuscript.
Author information
Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare no conflict of interest.
Additional information
Supplementary Information accompanies this paper on Heredity website
Supplementary information
Rights and permissions
About this article
Cite this article
Séré, M., Thévenon, S., Belem, A. et al. Comparison of different genetic distances to test isolation by distance between populations. Heredity 119, 55–63 (2017). https://doi.org/10.1038/hdy.2017.26
Received:
Revised:
Accepted:
Published:
Issue Date:
Further reading

Developing and quality testing of microsatellite loci for four species of Glossina
Infection, Genetics and Evolution (2020)

Do dental nonmetric traits actually work as proxies for neutral genomic data? Some answers from continental and globallevel analyses
American Journal of Physical Anthropology (2020)

Strong genetic structure among populations of the tick Ixodes ricinus across its range
Ticks and Tickborne Diseases (2020)

Alternative Quantifications of Landscape Complementation to Model Gene Flow in Banded Longhorn Beetles [Typocerus v. velutinus (Olivier)]
Frontiers in Genetics (2020)

Assessment of genetic diversity and genetic relationships of farm and laboratory quail populations in Japan using microsatellite DNA markers
Veterinary Medicine and Science (2020)