128
E. A. Thompson
pedigree and population data: gene mapping relies on having related individuals,
but the relationships do not need to be known.
The original version of this chapter was written in 2013–2016. HSG4, the 4th
edition of the Handbook of Statistical Genomics (Balding et al., 2019), was developed in 2017–2018. Thus, although the focus is slightly different, there is significant
overlap between the current chapter and Chapter 20 of HSG4 (Thompson, 2019).
6.2
Probabilities of IBD
6.2.1 IBD in Defined Relatives
In a defined pedigree relationship, the probabilities of IBD are determined by the
pedigree. As an example, we consider again the pair of cousins E and C in the
pedigree of Fig. 6.1. At a single locus, the probability of IBD of their maternal
gametes from any one of the four grandparental genes is (1/2) 4 = 1/16, since that
copy of the DNA must be chosen in each of four independent meioses. Of course,
E and C cannot share other than their maternal genomes IBD. Thus, in total, the
probability that E and C share DNA IBD at a locus is 4 × (1/16) or 1/4.
Suppose that, at a particular locus, DNA in one of the four genomes of the
shared grandparent couple does descend both to E and to C, then E and C do
share genome IBD at this locus. Moving along the chromosome, we assume, for
simplicity, that recombination in any meiosis occurs at random at an average rate
of 1 per 10 8 bp. Since a recombination event in any one of the four connecting
independent meioses will break this IBD, IBD is broken at a rate 4×10 −8 per bp, and
the expected distance until IBD is broken is 25 × 10 6 bp (25 Mbp), which is of order
25 centiMorgans (cM). That is, an IBD segment in first cousins has expected length
25 cM. Note however that this is different from the expected length of segment
surrounding a known IBD point, since the segment will extend in both directions
along the chromosome, for a total expected length of 50 cM. This apparent anomaly
is the phenomenon of size-biased sampling. It is well known in statistics (Cox,
1962) but has caused some confusion in the recent IBD literature. It is important to
distinguish between a randomly chosen IBD segment and the segment surrounding
a randomly chosen point of IBD.
Consider now the brothers C and D. At a locus they receive their maternal
DNA IBD with probability 1/2 and independently receive their paternal DNA IBD
with probability 1/2. Thus they share both homologues IBD with probability 1/4,
neither with probability 1/4. With the remaining probability 1/2, they share one of
their two homologues IBD. Since recombination in either of two meioses breaks a
maternal IBD segment, the expected length of such a segment is 50 cM and likewise
of a paternal IBD segment. The four meioses to the sibs from their parents are
independent, and every recombination in any one of the four parental meioses results
in a state change. Thus, along the chromosome, the IBD state remains constant for
an average of 25 cM. Switches in state occur from 1 to 0 or 2 IBD or from 0 or 2 to
1 IBD (Fig. 6.3).
E. A. Thompson
pedigree and population data: gene mapping relies on having related individuals,
but the relationships do not need to be known.
The original version of this chapter was written in 2013–2016. HSG4, the 4th
edition of the Handbook of Statistical Genomics (Balding et al., 2019), was developed in 2017–2018. Thus, although the focus is slightly different, there is significant
overlap between the current chapter and Chapter 20 of HSG4 (Thompson, 2019).
6.2
Probabilities of IBD
6.2.1 IBD in Defined Relatives
In a defined pedigree relationship, the probabilities of IBD are determined by the
pedigree. As an example, we consider again the pair of cousins E and C in the
pedigree of Fig. 6.1. At a single locus, the probability of IBD of their maternal
gametes from any one of the four grandparental genes is (1/2) 4 = 1/16, since that
copy of the DNA must be chosen in each of four independent meioses. Of course,
E and C cannot share other than their maternal genomes IBD. Thus, in total, the
probability that E and C share DNA IBD at a locus is 4 × (1/16) or 1/4.
Suppose that, at a particular locus, DNA in one of the four genomes of the
shared grandparent couple does descend both to E and to C, then E and C do
share genome IBD at this locus. Moving along the chromosome, we assume, for
simplicity, that recombination in any meiosis occurs at random at an average rate
of 1 per 10 8 bp. Since a recombination event in any one of the four connecting
independent meioses will break this IBD, IBD is broken at a rate 4×10 −8 per bp, and
the expected distance until IBD is broken is 25 × 10 6 bp (25 Mbp), which is of order
25 centiMorgans (cM). That is, an IBD segment in first cousins has expected length
25 cM. Note however that this is different from the expected length of segment
surrounding a known IBD point, since the segment will extend in both directions
along the chromosome, for a total expected length of 50 cM. This apparent anomaly
is the phenomenon of size-biased sampling. It is well known in statistics (Cox,
1962) but has caused some confusion in the recent IBD literature. It is important to
distinguish between a randomly chosen IBD segment and the segment surrounding
a randomly chosen point of IBD.
Consider now the brothers C and D. At a locus they receive their maternal
DNA IBD with probability 1/2 and independently receive their paternal DNA IBD
with probability 1/2. Thus they share both homologues IBD with probability 1/4,
neither with probability 1/4. With the remaining probability 1/2, they share one of
their two homologues IBD. Since recombination in either of two meioses breaks a
maternal IBD segment, the expected length of such a segment is 50 cM and likewise
of a paternal IBD segment. The four meioses to the sibs from their parents are
independent, and every recombination in any one of the four parental meioses results
in a state change. Thus, along the chromosome, the IBD state remains constant for
an average of 25 cM. Switches in state occur from 1 to 0 or 2 IBD or from 0 or 2 to
1 IBD (Fig. 6.3).