Runs of homozygosity and inbreeding
Long homozygous stretches mark chromosome segments inherited from one ancestor through both parents, and their lengths date the shared ancestor.
Read first: Genetic drift
One ancestor, two paths
When an individual's parents are related, a segment of the genome can come from the same ancestral chromosome down both sides of the family. The two copies in the child are then identical by descent (IBD), and the child is homozygous at every site in that segment. A stretch of this kind is a run of homozygosity (ROH).
The pedigree inbreeding coefficient F is the probability that the two copies at a locus are IBD. It is a sum over common ancestors: F = Σ (1/2)^(m − 1), where m is the number of meioses on the loop through the ancestor. For first cousins the loop has m = 6 and there are two grandparents, so F = 2 × (1/2)^5 = 1/16 = 0.0625. For second cousins m = 8 and F = 2 × (1/2)^7 = 1/64.
F is an expectation. What a person actually inherits is a fraction of the genome in IBD tracts, F_ROH, which varies from person to person. The module drops chromosome labels down the real pedigree with crossovers at every meiosis, lays SNP alleles on the founder segments, and calls ROH the way PLINK does. It then compares what it calls with the true IBD tracts.
First cousins, 40 genomes
Tract length dates the ancestor
Every meiosis on the loop can cut the tract with a crossover. Crossovers fall at one per Morgan per meiosis, so a tract ends at a rate of m per Morgan and its length is exponential with mean 100/m cM. For first cousins that is 100/6 = 16.7 cM. Chromosome ends cut tracts short, so the called mean runs a little below this in some scenarios and a little above in others once short tracts drop out.
The caller slides a window of 50 SNPs along each chromosome. A window is homozygous if it holds at most 1 heterozygous call. A SNP joins a run if at least 5% of the windows over it are homozygous. A run needs at least 50 SNPs and the minimum length, so the effective minimum is the larger of Min cM and 50 divided by the SNPs per cM. At the defaults, 25 SNPs per cM, that is 2 cM.
More meioses on the loop mean more cuts and shorter tracts. The ancestor of a second-cousin pair lived one generation further back than for first cousins, and the tracts are shorter by one quarter, 12.5 cM against 16.7. Tract length therefore tells how long ago the common ancestor lived, and the count of tracts tells how many ancestors there were.
Lengths follow the exponential
Exercise
Raise the minimum length
Open the module with the defaults and the seed 1. Set Min cM to 10 and run. Read the mean F_ROH and ROH per individual for First cousins, then switch the scenario to Small population.
What to expect. For first cousins the mean F_ROH falls from 0.058 to 0.049 and the ROH per individual from 11.1 to 6.2. Short tracts are no longer called, though they are still IBD. For the small population the mean F_ROH falls from 0.117 to 0.083 and the ROH per individual from 33.7 to 12.2, because its tracts are shorter still.
Open the walkthrough step →Same F, different tracts, and a wide spread
Three pedigrees share F = 1/8: half-sibs (m = 4), uncle and niece (m = 5), and double first cousins (m = 6 around two loops). Their expected tract lengths are 25, 20 and 16.7 cM, and children of each carry 17.4, 20.9 and 24.0 ROH on average. Two people with the same F can differ in how their inbreeding reads in the genome, and the shorter the tracts the more of them there are.
Realised F also varies within a scenario. Children of first cousins range from 0.006 to 0.112 in F_ROH at a mean of 0.0576. The standard deviation is 0.023, and theory gives 0.024. Second cousins show it more strongly: F = 1/64 = 0.0156, mean F_ROH 0.0154, 3.6 ROH per child, and some children have no ROH at all. Selfing (F = 1/2) gives a mean of 0.503 and 41 ROH of 40 cM on average, below the 50 cM expected because chromosome ends cut them.
The last scenario is a small isolated population of N = 50 founded 12 generations ago by unrelated individuals. Its F is 1 − (1 − 1/2N)^t = 0.114, similar to half-sibs, but the common ancestors are spread over many generations. Tracts are short (8.7 cM expected) and numerous (33.7 ROH per individual), and the spread of F_ROH is wide: 0.048 to 0.66, with a standard deviation of 0.097 where theory gives 0.077.
Realised F against pedigree F
Exercise
A smaller, older population
Select Small population, set N to 10 and t to 30, and run with the seed 1. Read the pedigree F and the ROH per individual.
What to expect. The pedigree F is 0.785, so most of the genome is IBD, and each individual carries 129 ROH. Tracts are cut by the crossovers of 30 generations, so the mean called length is 20.2 cM, inside a genome that is almost all homozygous.
Open the walkthrough step →Summary
- A ROH is a stretch inherited from one ancestor down both parents. Its fraction of the genome, F_ROH, is realised inbreeding, and pedigree F is only its expectation.
- Pedigree F is Σ (1/2)^(m − 1) over common ancestors: 1/16 for first cousins and 1/64 for second cousins.
- Tracts end at a rate of m per Morgan, so their mean length is 100/m cM: 16.7 cM for first cousins and 12.5 cM for second cousins.
- The minimum length is the larger of Min cM and 50 SNPs divided by the SNPs per cM. A higher minimum drops short tracts and lowers F_ROH.
- Same F does not mean the same tracts: half-sibs, uncle and niece and double first cousins (F = 1/8) differ in tract length and count.
- Realised F spreads widely, and most widely in small populations, where inbreeding comes from many generations at once.