The distribution of the fraction of the genome identical by descent in finite random mating populations

Abstract

SUMMAKYThe probability distribution of the heterogenic (non-identical by descent) fraction of the genome in a finite monoecious random mating population has been derived. It was assumed that in any generation the length of both heterogenic and homogenic segments are exponentially distributed. An explicit expression is given for the expected number of 'external junctions' (sites that mark the end of a heterogenic segment) per unit map length in any generation. The latter necessitates the introduction of two higher-order identity relations between three genes, and their recurrence relations. Theoretical results were compared with the outcome of a series of simulation runs (showing a very good fit), as well as with the results predicted by Fisher's 'theory of junctions'. In contrast to Fisher's approach, which only applies when the average heterogeneity is relatively small, the present model applies to any generation.

Marker densities and the mapping of ancestral junctions

“…Here we show that the expected number of junctions per Morgan in such a population is linearly related to the inbreeding coefficient of the population, with a maximum in a completely inbred population corresponding to the prediction given by Stam (1980). We further show that high-density marker maps (fully informative markers with average densities of up to 200 per cM) will fail to detect a significant proportion of the junctions present in highly inbred populations.…”

“…Junctions can be treated as point mutations, and followed to fixation or loss within the population over time. The genome of any individual in a partially inbred population will consist of alternating IBD and non-IBD sections, which are separated by 'external ' junctions (Stam 1980). In a completely inbred population, all individuals will be IBD across the whole genome, with each chromosome consisting of several segments of different lengths, each derived from a distinct chromosome in the base population, and separated by 'internal ' junctions (Stam, 1980).…”

“…The genome of any individual in a partially inbred population will consist of alternating IBD and non-IBD sections, which are separated by 'external ' junctions (Stam 1980). In a completely inbred population, all individuals will be IBD across the whole genome, with each chromosome consisting of several segments of different lengths, each derived from a distinct chromosome in the base population, and separated by 'internal ' junctions (Stam, 1980). Stam (1980), and Chapman & Thompson (2003) expand Fisher's theory by deriving expressions for the expectation and variance of the length of an IBD tract (the length of the section of chromosome between two external junctions), as opposed to the probability that an individual is IBD at a specific locus.…”

“…In a completely inbred population, all individuals will be IBD across the whole genome, with each chromosome consisting of several segments of different lengths, each derived from a distinct chromosome in the base population, and separated by 'internal ' junctions (Stam, 1980). Stam (1980), and Chapman & Thompson (2003) expand Fisher's theory by deriving expressions for the expectation and variance of the length of an IBD tract (the length of the section of chromosome between two external junctions), as opposed to the probability that an individual is IBD at a specific locus. The papers of Stam and Chapman & Thompson relate to random mating populations excluding and including selfing, respectively.…”

“…The papers of Stam and Chapman & Thompson relate to random mating populations excluding and including selfing, respectively. Stam (1980) showed that starting with a fixed outbred population of size N, with a genome of length L Morgan distributed over n pairs of homologous chromosomes, and allowing the population to mate randomly with the exception of selfing, until the population was entirely inbred, the number of distinct segments observed in an individual in the inbred population (S O ) would be :…”