PROPERTIES OF Δ
For full explanation see: Gillet EM, Gregorius H-R. 2008. Measuring differentiation among populations at different levels of genetic integration. BMC Genetics 9, 60. http://dx.doi.org/10.1186/1471-2156-9-60
For a given set of gene loci, the following always holds for the genetic distances Δ(P,Q) between any two populations P and Q at the different levels of genetic integration:
- The mean over loci of the distances Δ(P,Q) between the allelic frequency distributions (genepool) cannot be greater than the mean over loci of the distances Δ(P,Q) between the frequency distributions of the single-locus genotypes.
- The mean over loci of the distances Δ(P,Q) between the frequency distributions of the single-locus genotypes cannot be greater than the distance Δ(P,Q) between the frequency distributions of the multilocus genotypes.
Some special cases in which Δ(P,Q) does not increase from a lower to a higher level of genetic integration:
Case 1: For a given gene locus, let the genotypes of the individuals in population P and population Q show Hardy-Weinberg proportions. Then the following measures are equal:
- Δ(P,Q) at the integration level of the single-locus genotypes
- Δ(P,Q) at the integration level of the genepool (alleles)
- d0(P,Q) at the level of the genepool.
The same equality holds if P and Q both show inbreeding structures with the same value of the inbreeding coefficient F (Hardy-Weinberg proportions correspond to inbreeding structures with F=0.)
Case 2: For a given set of gene loci, let the multilocus genotypes of the individuals in population P and in population Q show random association of the single-locus genotypes among gene loci . Then the following measures are equal:
- Δ(P,Q) at the integration level of the multilocus genotypes
- Δ(P,Q) at the integration level of the mean single-locus genotypes
Combination of Cases 1 and 2: If, for a given set of gene loci, population P and population Q show Hardy-Weinberg proportions (or equal inbreeding coefficients F) at all loci as well as random association of the single-locus genotypes among all loci, then the following measures are equal:
- Δ(P,Q) at the integration level of the multilocus genotypes
- Δ(P,Q) at the integration level of the mean single locus genotypes
- d0 at the integration level of the genepool.