partial_genotype_order {gpmap}

Generate partial ordering of genotype space based from allele content of genotypes

Description

Generate the strict partial order on genotype space specified in eq. (13) in Gjuvsland et al. (2011). For a genotype space with N biallelic loci a minimal description of this partial order is given by 2N*3^{(N-1)} inequalities.

Usage

 partial_genotype_order(plusallele) 

Arguments

plusallele

A N-vector with indexes (1 or 2) for the +allele for each locus

Details

In short a partial order on a set is a binary relation defining a parwise ordering of some pairs of elements in the set, for a formal definiton see https://en.wikipedia.org/wiki/Partial_order. In the partial order on the set of genotypes defined in Gjuvsland et al. (2011) the comparable pairs of genotypes are equal at every locus except one, while all other pairs of genotypes are incomparable. This partial ordering of genotype space is implicit in the regression on gene content (the number of alleles with a given index in each genotype) used for decomposition of tne genotypic value in quantitative genetics (see e.g. Lynch and Walsh page 65).

Value

Returns a (2N*3^{(N-1)}) \times 2 matrix of genotype indexes. The genotype indexes refer to row number in the genotype sequence set up in enumerate_genotypes. Each row vector in the matrix contains the genotype indexes of one comparable pairs, and if the first index is k and the second is l then genotype[k] < genotype[l].

Author(s)

Arne B. Gjuvsland <[e-mail]> and Yunpeng Wang <[e-mail]>

References

Gjuvsland AB, Vik JO, Woolliams JA, Omholt SW (2011) Order-preserving principles underlying genotype-phenotype maps ensure high additive proportions of genetic variance. Journal of Evolutionary Biology 24(10):2269-2279 [link]

Lynch M & Walsh B (1998) Genetics and Analysis of Quantitative Traits, Sunderland, MA: Sinauer Associates