Variogram analysis of the spatial genetic structure of continuous populations using multilocus microsatellite data

Abstract

A geostatistical perspective on spatial genetic structure may explain methodological issues of quantifying spatial genetic structure and suggest new approaches to addressing them. We use a variogram approach to (i) derive a spatial partitioning of molecular variance, gene diversity, and genotypic diversity for microsatellite data under the infinite allele model (IAM) and the stepwise mutation model (SMM), (ii) develop a weighting of sampling units to reflect ploidy levels or multiple sampling of genets, and (iii) show how variograms summarize the spatial genetic structure within a population under isolation-by-distance. The methods are illustrated with data from a population of the epiphytic lichen Lobaria pulmonaria, using six microsatellite markers. Variogram-based analysis not only avoids bias due to the underestimation of population variance in the presence of spatial autocorrelation, but also provides estimates of population genetic diversity and the degree and extent of spatial genetic structure accounting for autocorrelation.

Figure 1.—

Empirical variogram (A) and correlograms (B) for an artificial, spatially autocorrelated random variable simulated on a grid of 30 × 30 cells. Each symbol denotes the semivariance γ(r) (circles), Geary's c(r) (squares), or Moran's I(r) (triangles) calculated from all pairs of samples falling into each distance class r. The value for the last distance class of each series contains all pairs separated by >20 units and is drawn at the mean of the respective distances. (A) The solid line represents the fitted exponential variogram model. The dotted line (sill) indicates the population variance as estimated accounting for autocorrelation. The dashed line indicates the practical range, where the curve reaches 95% of the sill. The intercept (nugget variance) is the variance component that is not spatially structured. (B) The solid line indicates the expected value of Moran's I(r), which is very close to zero, whereas the dashed line marks the expected value of Geary's c(r), which equals one.

Figure 2.—

Variogram of gene diversity Ĥ(r) for a population of Lobaria pulmonaria. Each symbol denotes the mean semivariance over six microsatellite loci averaged over all pairs of thalli within each distance class. The semivariance is unweighted (circles) or weighted for recurrent genotypes (squares). The lines indicate the corresponding fitted exponential models; the dashed line shows the nonspatial model. Values below the dashed line correspond to positive, and values above the dashed line to negative, autocorrelation. Solid symbols indicate statistically significant positive autocorrelation based on a one-sided Mantel permutation test with progressive Bonferroni correction (α = 0.05).