Figure 3
From: Haplotype inference from unphased SNP data in heterozygous polyploids based on SAT
Usage of Boolean variables. Panel A shows a population of three unphased genotypes. Each of the tetraploid biallelic genotypes consists of two SNP sites. Variable names and usage of indices are given. Colour of genotype represents the coloured mixture of the explaining haplotypes. Yellow variables represent selection variables. In the model, phased genotypes are represented by selection variables. However, the possible phase of genotypes (explanation by haplotypes) is also shown. Panel B to G are read from left to right and top to bottom. If a genotype can be explained by four haplotypes from a given a set of haplotypes, it is tried to explain the next genotype in the population. Panel B and C: one haplotype is assumed as most parsimonious set and it is tried to explain genotype g1. Above the dashed line, the assumed r haplotypes and all four possible most parsimonious sets are shown. Below the dashed line, unphased genotype g1 and each try to explain genotype g1 by the assumed set of haplotypes is given. If no satisfying assignment can be found, corresponding cells are not filled with Boolean values. Relationships between explaining haplotypes and the corresponding column of selection variables is symbolised by diagonal lines. Rows of selection variables are constrained to sum up to 1 since exactly one haplotype is to be selected for representing a chromosome. Panel D and E: two haplotypes are assumed as most parsimonious set. All possible sets of haplotypes are shown. Strict lexicographic order breaks symmetries in sets of explaining haplotypes. The third and fourth set of explaining haplotypes can explain genotype g1 but fail to explain genotype g2. Panel F and G: three haplotypes are assumed as most parsimonious set. The first set is not able to explain the whole population. In contrast, the second set is able to explain all three genotypes. Thus, this set solves the HIPP for the given population.