Plotting kinship using IBS relationship matrix data (Supplement, Seroussi et al., 2012).

Using whole genome SNP data, PLINK software offers a powerful and simple approach to population stratification. This includes genome-wide identity-by-state (IBS) counts for all pairs of individuals. Each SNP locus presents 3 possible states: none of the alleles matches (IBS0), one allele match (IBS1), and both alleles match (IBS2). For each pair, the frequency of occurrence of these states (freq1, 2 and 3, respectively) can be estimated by dividing these counts by the total number of loci for which genotypes were determined in both individuals. Clustering of pairs according to the degree of kinship is clearly revealed when plotting these results in three dimensions (3D) (Fig. S1).


Fig. S1Key


Fig. S1. Kinship within the population of Israeli artificial insemination (AI) bulls- 3D plot. Using SNP50 BeadChip data of 789 sires, the genome-wide identity-by-state of each possible pair of individuals is illustrated by plotting a colored dot at the 3D coordinates that correspond to the frequency of the 3 possible states of the SNPs’ alleles (freq0- no match, freq1- single match, freq2- both match). Dot colors indicate the familial relation and kinship level according to the following key.


As the sum of the frequencies of the 3 possible states is always one, the points are always located on a single plane that intersects the axes at value of unity (Fig. S2). This plane’s intersections with the x,y, xz and xz planes form an equilateral triangle (side length √2). The inclination of this triangle to the planes is a constant (54.7o).

Fig. S2


Fig. S2. Cartesian coordinates on kinship data plane. Axes (X’, Y’) were fitted to the plane formed by the frequency of the 3 possible states of the SNPs’ alleles for the data described in Fig. S1. The point P has coordinates x, y and z (black) which correspond to coordinates x’, y’ on the fitted axes (red).


The 3D coordinates of any point (P) can be transformed to 2D coordinates on the axes fitted to the kinship plane (Fig. S2). Since the altitude (d, Fig. S2) is equal in both right-angled triangles whose base sides form the X’ axe, the equation (Eq. 1) can be phrased according to the Pythagorean Theorem:

Eq. 1    d2=z2+(1-y)2-(√2-x’)2=y2+(1-z)2-x’2

z2+1-2y+y2-2+2√2x’-x’2=y2+1-2z+z2-x’2

2√2x’-x’2+x’2=y2+1-2z+z2-z2-1+2y-y2+2

2√2x’=2-2z+2y=2(1-z+y)

                                    Eq. 2    x’=0.7071(1+y-z)

                                    Eq. 3    x’=0.7071(1+freq1-freq2)

Y’ is inferred from the right triangle with base d and altitude x (Fig. S2) using the sine function.

                                    Eq. 4    y’=x/sin(54.7o)=1.2247x

                                    Eq. 5    y’=1.2247freq0

Application of equations 3 and 5 to the BeadChip 3D data of the Israeli AI sires (Fig. S1) projects this kinship illustration onto a bi-dimensional plain (Fig. S3).


Fig. S3
Key2

Fig. S3. Kinship within the population of Israeli artificial insemination (AI) bulls- 2D plot. Using SNP50 BeadChip data of 789 sires, the genome-wide identity-by-state of each possible pair of individuals is illustrated by plotting a colored dot at the 2D coordinates (X’, Y’) that correspond to the frequency of the 3 possible states of the SNPs’ alleles (freq0- no match, freq1- single match, freq2- both match) using the transformation x’=0.7071(1+freq1-freq2) and y’=1.2247freq0. Dot colors indicate the familial relation and kinship level according to the following key. Dots representing relation to the maternal grandsires are double sized (red).