At the individual level, the pattern of spatial genetic structure was explored by spatial autocorrelation methods (Vekemans & Hardy, 2004) on the 124 individuals over the continuous population. The association between pairwise relationship coefficients calculated on AFLP markers (Hardy, 2003) in spagedi (Hardy & Vekemans, 2002) and the logarithm of the geographic distance was tested by Mantel tests with 9999 permutations, using fstat (Goudet, 1995). An autocorrelogram was then constructed to assess correlations among genotypes at increasing geographical distance intervals with balanced number of individuals. Significance of distance-class mean relationship coefficients was assessed with 999 permutations and Bonferroni corrected (Vekemans & Hardy, 2004).
Given the lack of knowledge concerning the segregation of dominant AFLP markers in the autopolyploid B. laevigata (i.e. disomic vs multisomic inheritance, rate of double reduction), the genetic structure of the continuous population was investigated by a multivariate analysis (Fig. 2). Since it is a band-based rather than allele frequency-based approach, this procedure does not assume Hardy–Weinberg equilibrium and is therefore convenient to explore the genetic structure of polyploid populations with multisomic inheritance. Using ADE-4 (Thioulouse et al., 1997), PCA on the covariance matrix was computed on AFLP profiles at the individual level and then at the plot level with BPCA in order to subsequently correlate genetic data with environmental factors. Principal component analysis conserves Euclidian distances and decomposes the covariance of all descriptors (here, loci) into components for each object (here, individuals or plots) along each of the full-ranked eigenvectors derived from general singular value decomposition (Doledec & Chessel, 1987; Patterson et al., 2006). Principal component analysis thus summarizes a maximum of variance into fewer, interpretable dimensions. Thereafter, BPCA (i.e. PCA between plots based on PCA among individuals) was performed. This analysis groups individual PCA profiles into sampling plots in order to maximize the between-group genetic variance. The plot-centroids are then projected in a new reduced space, along the full-ranked BPCA eigenvectors. Significance of the between-group variance was estimated by 9999 permutations using ADE-4 (Thioulouse et al., 1997). The mathematical details of the BPCA can be found elsewhere (Culhane et al., 2002; Pavoine et al., 2004), but variance partitioning by BPCA is an Euclidian discriminant approach and can be safely used with any combination of plots and loci. One interesting use of BPCA is to produce a set of univariate genotypic variables for each plot (BPCA scores) that can be further analysed by univariate and/or spatial statistics in order to help the interpretation of CCA (see later). Since BPCA is based on Euclidian distances, it can be considered as analogous to F-statistics (Parisod et al., 2005). However, those estimators, named βST, are not equivalent to F-statistics and βST values may be overestimated because BPCA maximizes the between-group variance. Nevertheless, relative βST represented the genetic structure of the population well. Indeed, pairwise βST, which were computed here as multidimensional Euclidian distance between the multidimensional BPCA scores of plot centroids, were slightly inflated but highly correlated with other traditional estimators of genetic differentiation (Supplementary material, Fig. S1), such as the GST calculated on Shannon diversity (Mantel test, r = 0.81, P-value < 0.001; Bussell, 1999) and the ΦST obtained from the amova implemented in the arlequin software (Mantel test, r = 0.70, P-value < 0.001; Excoffier et al., 1992).
Figure 2. Conceptual diagram of the multivariate statistical procedure used to investigate the association of individual genetic amplified fragment length polymorphism (AFLP) profiles and environmental factors. Variance partitioning by between-group eigenanalyses (BPCA) is analogous to FST and can be used to summarize most of the individual genetic information in a subspace at the plot level. Similarly, most of the environmental information is combined into an environmental subspace at the plot level. i and n represent the number of individuals and plots, respectively. j and p are the number of AFLP loci and environmental factors, respectively. After multivariate procedures (presented along the arrows), the column of the matrix contains independent information (scores, j′ or j″ and p′) that can be further investigated by univariate and/or spatial analyses. The canonical correspondence analysis (CCA, at the bottom) provides the framework to regress the multivariate environmental subspace on the multivariate genetic subspace and test for their association by permutations. Euclidian distances between plots multidimensional scores (j″ and p′) can be computed in each subspace (dashed arrows) and used to investigate the influence of the geographical distance between plots with partial Mantel tests (at the top).
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The relationships between the Shannon diversity index, plot size and density, as well as ecological factors were explored by stepwise multiple linear regressions and robust regressions for each factor.