130
E. A. Thompson
the rates of moving out of a given IBD state are no longer the same. C and D will
share their maternal genomes for an average length of 50 cM, but IBD with E will be
more rapidly broken, because of the greater number of intervening meioses. Also,
it is no longer sufficient to consider only the rate of breaking an IBD chain. For
example, E m can switch directly from being IBD with C m to being IBD with D m .
Despite the rapidly increasing complexity of IBD states as more individuals are
considered, the specification of inheritance in a defined pedigree is straightforward.
Suppose all the meioses of a pedigree are indexed by m, m = 1, . . . , M; M is twice
the number of non-founders in the pedigree, since each non-founder has a maternal
and a paternal meiosis giving rise to their maternal and paternal gametes. Suppose
L locations of interest across a chromosome are indexed by loci j , j = 1, . . . , L.
We define, for each meiosis m and location j
S mj = 1 if the parent’s paternal DNA is transmitted
S mj = 0 if the parent’s maternal DNA is transmitted
(6.1)
Then Mendel’s first law states that meioses m are independent and that
Pr(S mj = 1) = Pr(S mj = 0) = 1/2.
(6.2)
Secondly, under the assumption of no genetic interference (Haldane, 1919), the
crossover points in the gametes transmitted to offspring (Fig. 6.2) occur independently and at random at rate 0.01 per cM. Then the {S mj ; j = 1, . . . , L} have a
Markov dependence over j . This can be expressed as
Pr(S mj = 1 | S m ,j , (m
, j
) = (m.j )) = Pr(S mj = 1 | S m,(j −1) , S m,(j +1) )
(6.3)
That is, given all the other S m j , S mj depends only on the values S m,(j −1) and
S m,(j +1) for the same meiosis m and the two neighboring loci. The vector of
components S mj over the values of m for any given locus j is known as the
inheritance vector at locus j (Lander and Green, 1987).
Equations (6.2) and (6.3) provide easy methods for simulation of the descent
of genome in a defined pedigree. At independently inherited loci, it is simply
an application of Mendel’s first law, with the parents maternal or paternal DNA
each being transmitted independently in every meiosis. Across the genome, the
copying switches between copying from the parent’s maternal and paternal DNA
at a rate determined by the genetic map. Under the model of no genetic interference
(Haldane, 1919), the distance to the next switch point on each chromosome can be
generated as an exponential random variable with mean 100 cM. The values of S mj
are then determined at the specified discrete locations j . For each j , or indeed jointly
over j , the S mj determine which founder genome descends to each haploid genome
of each current individual and hence the IBD state among current individuals. Thus
Monte Carlo estimates of the probabilities of IBD patterns, both at a locus and across
loci, can be very efficiently obtained.
E. A. Thompson
the rates of moving out of a given IBD state are no longer the same. C and D will
share their maternal genomes for an average length of 50 cM, but IBD with E will be
more rapidly broken, because of the greater number of intervening meioses. Also,
it is no longer sufficient to consider only the rate of breaking an IBD chain. For
example, E m can switch directly from being IBD with C m to being IBD with D m .
Despite the rapidly increasing complexity of IBD states as more individuals are
considered, the specification of inheritance in a defined pedigree is straightforward.
Suppose all the meioses of a pedigree are indexed by m, m = 1, . . . , M; M is twice
the number of non-founders in the pedigree, since each non-founder has a maternal
and a paternal meiosis giving rise to their maternal and paternal gametes. Suppose
L locations of interest across a chromosome are indexed by loci j , j = 1, . . . , L.
We define, for each meiosis m and location j
S mj = 1 if the parent’s paternal DNA is transmitted
S mj = 0 if the parent’s maternal DNA is transmitted
(6.1)
Then Mendel’s first law states that meioses m are independent and that
Pr(S mj = 1) = Pr(S mj = 0) = 1/2.
(6.2)
Secondly, under the assumption of no genetic interference (Haldane, 1919), the
crossover points in the gametes transmitted to offspring (Fig. 6.2) occur independently and at random at rate 0.01 per cM. Then the {S mj ; j = 1, . . . , L} have a
Markov dependence over j . This can be expressed as
Pr(S mj = 1 | S m ,j , (m
, j
) = (m.j )) = Pr(S mj = 1 | S m,(j −1) , S m,(j +1) )
(6.3)
That is, given all the other S m j , S mj depends only on the values S m,(j −1) and
S m,(j +1) for the same meiosis m and the two neighboring loci. The vector of
components S mj over the values of m for any given locus j is known as the
inheritance vector at locus j (Lander and Green, 1987).
Equations (6.2) and (6.3) provide easy methods for simulation of the descent
of genome in a defined pedigree. At independently inherited loci, it is simply
an application of Mendel’s first law, with the parents maternal or paternal DNA
each being transmitted independently in every meiosis. Across the genome, the
copying switches between copying from the parent’s maternal and paternal DNA
at a rate determined by the genetic map. Under the model of no genetic interference
(Haldane, 1919), the distance to the next switch point on each chromosome can be
generated as an exponential random variable with mean 100 cM. The values of S mj
are then determined at the specified discrete locations j . For each j , or indeed jointly
over j , the S mj determine which founder genome descends to each haploid genome
of each current individual and hence the IBD state among current individuals. Thus
Monte Carlo estimates of the probabilities of IBD patterns, both at a locus and across
loci, can be very efficiently obtained.