Ancestral Relationships Using Metafounders: Finite Ancestral Populations and Across Population Relationships

Abstract

Recent use of genomic (marker-based) relationships shows that relationships exist within and across base population (breeds or lines). However, current treatment of pedigree relationships is unable to consider relationships within or across base populations, although such relationships must exist due to finite size of the ancestral population and connections between populations. This complicates the conciliation of both approaches and, in particular, combining pedigree with genomic relationships. We present a coherent theoretical framework to consider base population in pedigree relationships. We suggest a conceptual framework that considers each ancestral population as a finite-sized pool of gametes. This generates across-individual relationships and contrasts with the classical view which each population is considered as an infinite, unrelated pool. Several ancestral populations may be connected and therefore related. Each ancestral population can be represented as a "metafounder," a pseudo-individual included as founder of the pedigree and similar to an "unknown parent group." Metafounders have self- and across relationships according to a set of parameters, which measure ancestral relationships, i.e., homozygozities within populations and relationships across populations. These parameters can be estimated from existing pedigree and marker genotypes using maximum likelihood or a method based on summary statistics, for arbitrarily complex pedigrees. Equivalences of genetic variance and variance components between the classical and this new parameterization are shown. Segregation variance on crosses of populations is modeled. Efficient algorithms for computation of relationship matrices, their inverses, and inbreeding coefficients are presented. Use of metafounders leads to compatibility of genomic and pedigree relationship matrices and to simple computing algorithms. Examples and code are given.

Keywords: GenPred; base populations; genetic drift; marker genotypes; pedigree; relationships; shared data resource.

Figure 1

Ancestral and base population and pedigree.

Figure 2

Relationships (red dashed line), in a related base population, of two individuals X and Y seen as gametes (left) or individuals (right). Individual X (Y) contains gametes a and b (c and d). Relationships across and within gametes are respectively $\mathit{ }\gamma /2$ and 1; relationships across and within individuals are γ and $1+\gamma /2$.

Figure 3

Base population with a metafounder and corresponding pedigree.

Figure 4

Several related base populations.

Figure 5

Population with two related metafounders 1 and 2, self-relationship coefficients ${\gamma }^1$, ${\gamma }^2$, and relationship coefficient ${\gamma }^{1,2}$ and associated pedigree.

Figure 6

Inverse of the numerator relationship matrix with (up) unknown parent groups (equivalently, with Γ= 0) or (down) with metafounders and Holstein–Jersey Γ coefficients () scaled to the same genetic variance. Pedigree as in Figure 5.