Genetic contributions of founders and ancestors

Coancestry or kinship (A) matrices were estimated for individuals within continent-, region-, and country-specific focal populations using the create.pedigree and kin functions in the R package synbreed (version 0.12-9; Wimmer et al. 2012). Focal populations consisted of cultivars and their ascendants (ancestors). Founders are ancestors with unknown parents, which were assumed to be unrelated (Lacy 1989, 1995; Hartl and Clark 2007), whereas non-founders are ancestors with known parents. Terminal nodes in a pedigree network (sociogram) are either founders or the youngest descendants. The mean kinship (MK) between the ith founder and cultivars in a focal population was estimated by

where C_ij = the kinship coefficient between the ith founder and jth cultivar in a focal population, i=1,2,…,n, j=1,2,…,k, n = the number of founders in the focal population, and k = the number of cultivars in the focal population (Lacy 1989, 1995; Lynch and Walsh 1998; Hartl and Clark 2007). The proportional GC of the ith founder to a focal population was estimated by Pi=MKi/∑iMKi. The number of founder equivalents (F_e) was estimated by Fe=1/∑iMKi, where i∈{founder1,founder2,…,foundern} (Lacy 1989, 1995). Founder equivalents “are the number of equally contributing founders that would be expected to produce the same genetic diversity as in the population under study” (Lacy 1989).

The GCs of ancestors (founders and non-founders) to a focal population were estimated by constructing a directed distance matrix (D) with dimensions identical to A (n × n) such that parents appeared in the matrix before offspring (alleles flow from parents to offspring, but not vice versa). We used the directed distance (the number of PO edges between two accessions) to modify A so that coancestry coefficients were only estimated between ancestors and direct path cultivars. The directed distance matrix D was estimated using the distances function in the R package igraph (version 1.2.5; Csardi and Nepusz 2006), where nonzero distances in the D matrix were set equal to one. Coancestry coefficients for ascendants with no direct path to a cultivar were set equal to zero by taking the Hadamard product to generate the corrected coancestry matrix A⋆=A⊙D, where element C_ii = the coancestry coefficient for individual i with itself (Hartl and Clark 2007). To estimate GC for each ancestor, we applied an iterative approach that entailed (1) computing D, A, and A⋆=A⊙D from the current pedigree; (2) estimating MK_i for each ancestor; (3) ranking MK_i estimates from largest to smallest; (4) setting GC_i = MK_i for the ancestor with the largest MK_i estimate; (5) deleting the ancestor with the largest MK_i estimate and rebuilding the pedigree; and (6) repeating the previous steps until GCs (GC_i) had been estimated for each ancestor. The proportional GC of the ith ancestor to a focal population was estimated by Pi=GCi/∑iGCi.