Analysis of allele-frequency clines
We conducted three statistical tests for the occurrence of allele-frequency clines, which are explained in detail in the following paragraphs. First, we estimated the significance of regression of average allele frequency across loci against latitude. Second, we tested whether we could attribute the regressions (clines) to selection. Finally, we tested whether selection resulted in increased allele frequencies across the entire latitudinal range. In these tests, we excluded HiP, because here the male-deleterious alleles have earlier been shown to be under strong negative selection (van Hooft et al., 2019). In all tests, we pooled the frequencies of male-deleterious-trait-associated alleles at each locus (Table S1). We refer to the pooled alleles as DE and SAE alleles. Allele-frequency clines were based on plotting the DE and SAE allele frequencies as a function of latitude. The results of the three tests were not meaningfully influenced by sample-size weights, potential errors in allele frequency correction (see next paragraph), or by allele size standardization (Text S2).
In the first test (regression of average allele frequency against latitude), we corrected the average allele frequencies across microsatellites to account for the fact that populations were analysed with different microsatellite subsets using KNP frequencies as a standard for this correction. Specifically, we first calculated the correction factor for the i th population in terms of the relative frequencies of the alleles found in populationi compared with the full microsatellite set of 17 alleles found in both the northern (NK) and southern (SK) Kruger populations. If we use the notation \({\overset{\overline{}}{f}}^{\text{NK}}\) and\({\overset{\overline{}}{f}}^{\text{SK}}\) to represent the average frequencies of the 17 alleles found in northern and southern KNP and\({{\overset{\overline{}}{f}}_{i}}^{\text{NK}}\) and\({{\overset{\overline{}}{f}}_{i}}^{\text{SK}}\) to represent the averages in northern and southern KNP over only those alleles found in population i (some subset of the 17 alleles found in KNP), then our corrected average frequency\({{\overset{\overline{}}{f}}_{i}}^{\text{cor}}\) for populationi , in terms of the observed average frequency\({{\overset{\overline{}}{f}}_{i}}^{\text{obs}}\) of the subset of alleles in population i is given by:
\({{\overset{\overline{}}{f}}_{i}}^{\text{cor}}={{\overset{\overline{}}{f}}_{i}}^{\text{obs}}({\overset{\overline{}}{f}}^{\text{NK}}+{\overset{\overline{}}{f}}^{\text{SK}})/({{\overset{\overline{}}{f}}_{i}}^{\text{NK}}+{{\overset{\overline{}}{f}}_{i}}^{\text{SK}})\)(1)
This correction assumes a similar frequency ratio in all populations. The ratios were estimated for all loci combined, and for DE and SAE loci separately. The frequency correction had only a minor effect as the ratios varied by no more than a factor of 1.1 (range: 0.92-1.09), except for the Caprivi Strip population for which data from only two loci were available.
The large variation in number of genotyped individuals and number of genotyped loci between populations resulted in significant heteroskedasticity in our regression models (modified Breusch-Pagan test, Text S3) (Wooldridge, 2013). Correction for heteroskedasticity is possible by weighing each population by the number of genotyped individuals multiplied by the average number of genotyped loci per individual, because the standard deviations of the error terms are expected to scale linearly with the square-root of ‘within-group sample size’. However, we took a slightly different approach by weighing the regressions (of average allele frequency across loci against latitude) by the sum of the square-roots of the number of genotyped individuals per population (i.e., by \(\sum_{l=1}^{n}\sqrt{g_{l}}\), where\(g_{l}\) is the number of genotyped individuals at locus l , withl = 1,…,17). This is more appropriate here instead of the square-root of the sums because it gives more weight to the number of genotyped loci rather than to the number of genotyped individuals. In this way, relatively low weight was given to the Caprivi Strip population despite its large sample size (two genotyped loci, 134 genotyped individuals). Also, we weighted by square-root per locus, because the adjusted R 2 values were strongly positively biased by the relatively large sample sizes of the two KNP subpopulations. Besides significance in the Breusch-Pagan test, the effect of heteroskedasticity was also evident from the increased adjusted R 2 values relative to the unweighted regressions. The regressions were conducted with SPSS 23.
As a control, we also estimated the allele-frequency clines of the remaining alleles from southern KNP observed ≥15 times that were not associated with male-deleterious traits, again after pooling frequencies of individual alleles. We based these control clines on the alleles that were closest in size (number of short tandem repeats) to the non-pooled male-deleterious-trait associated alleles. Further, where possible, we selected the same number of non-pooled alleles per locus as with the male-deleterious-trait-associated alleles (Table S1). Only at three loci was the number of selected remaining control alleles smaller than the comparison alleles (BM3517 and TGLA227 : one instead of two alleles; BM1824 : two instead of four alleles).
In constructing the allele-frequency clines, we used an updated definition of DE and SAE alleles according to a recent genetic study in HiP (i.e., pooling all alleles with a relatively high frequency in BTB-positive males with low body condition) (van Hooft et al., 2019), rather than the one applied for the allele-frequency clines in KNP (van Hooft et al., 2014). Also with the new definition, a significant cline was observed in KNP for DE and SAE alleles combined and for DE alleles specifically, but not for SAE alleles (DE loci: adjustedR 2 = 0.23, P = 0.0042; SAE loci: adjusted R 2 = 0.04, P = 0.15; all loci: adjusted R 2 = 0.23, P = 0.0040; frequency against latitude, weighted by square-root of number of sampled individuals per herd; Figures S2 and S3). However, the Spearman correlation between average SAE allele frequency and latitude, when weighted by square-root of sample size, was close to significance (ρ = -0.36, P = 0.054).
In the second test (attribution of allele-frequency clines to selection), we estimated the significance of the average per-locus Pearson correlation between allele frequency and latitude, with the individual per-locus correlations weighted by the square-root of the number of genotypes per population. In the third test (increased allele frequencies across whole latitudinal range), we estimated whether average frequencies per locus were significantly higher for the DE and SAE alleles than for the control alleles, with the per-locus frequencies weighted by the square-root of the number of genotypes per population. In both tests, randomized pooled frequencies were obtained by replacing the non-pooled male-deleterious-trait-associated alleles (as mentioned before, DE and SAE alleles consisted of pools of male-deleterious-trait-associated alleles) with a random selection of non-pooled alleles (73 alleles observed ≥15 times in southern KNP, consisting of 33 male-deleterious-trait-associated alleles and 40 remaining alleles). We estimated significance as the probability that a random selection of non-pooled alleles resulted in a stronger average Pearson correlation per locus (Test 2) or a higher average frequency per locus (Test 3) than the male-deleterious-trait-associated alleles. We applied 100,000 randomizations implemented using Excel 2016.