Thore Egeland

1
Descent graphs in pedigree analysis applications
to haplotyping, location score, and
marker-sharing statistics.

• Thore Egeland

Based partly on notes by Maayan Fishelson
andTerry Speed
2
The general parts of the title

• - haplotyping
• - location score
• - marker-sharing statistics

3
References

• The algorithm presented herein was introduced by
• Sobel and Lange 2, and Kruglyak et al. 1.
• E. Sobel and K. Lange. Descent graphs in pedigree
  analysis applications to haplotyping, location
  score, and marker-sharing statistics. Am. J. Hum.
  Genet., 581323--1337. 1996.
• L. Kruglyak, M.J. Daly, M.P. Reeve-Daly, and E.S.
  Lander. Parametric and nonparametric linkage
  analysis a unified multipoint approach. Am. J.
  Hum. Genet., 581347--1363, 1996.

4
Inheritance vector I
1
2
(x1,x2)
(x3,x4)
paternal
f2 founders, n2 non-founders
12
13
(x1,x3)
(x2,x3)
v(x)(p1,m1,.,pn,mn)(1,1,0,1)
inheritance vector (Notation differs, I stick to
Fishelson, Speed)
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Inheritance vector II

• - All 22n inheritance vectors are equally
  likely a priori
• - Number of possible inheritance vectors reduced
  as people are genotyped.
• - If all are genotyped and phase is known, there
  is only one possible inheritance vector

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Descent Graph

• Corresponds to a specific inheritance vector.
• Sobel Lange construct Markov Chain on descent
  graphs
• Simulated annealing used to search for single
  most likely descent graph

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Main Idea

• Let a (a1,,a2f) be a vector of alleles
  assigned to founders of the pedigree (f is the
  number of founders).
• We want to represent by a graph the restrictions
  imposed by the observed marker genotypes on the
  vectors a that can be assigned to the founder
  alleles.
• The algorithm extracts from the graph only
  vectors a compatible with the marker data.

8
Example marker data on a pedigree
9
Descent Graph

• Corresponds to a specific inheritance vector.
• Vertices the individuals alleles (2 alleles for
  each individual in the pedigree).
• Edges represent the allele flow specified by the
  inheritance vector. A childs allele is
  connected by an edge to the parents gene from
  which it flowed.

10
Example Descent Graph (vertices)
Assume that the descent graph vertices below
represent the pedigree on the left.
Descent Graph
3
4
5
6
1
2
7
8
(a,b)
(a,b)
(a,c)
(b,d)
(a,b)
(a,b)
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Example Descent Graph (cont.)
Descent Graph
3
4
5
6
1
2
7
8
(a,b)
(a,b)
(a,c)
(b,d)
(a,b)
(a,b)

• Assume that paternally inherited alleles are on
  the left.
• Assume that non-founders are placed in
  increasing order.
• A 1 (0) is used to denote a paternally
  (maternally) originated gene.
• ? The gene flow above corresponds to the
  inheritance vector v ( 1,1 0,0 1,1 1,1
  1,1 0,0 )

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Founder Graph

• Vertices the founder alleles.
• Edges connect the alleles appearing together in
  a genotyped individual for the gene flow
  specified by the inheritance vector v.
• Note the edges are labeled with the genotype of
  the corresponding individuals.

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Example Founder Graph
Descent Graph
3
4
5
6
1
2
7
8
(a,b)
(a,b)
(a,c)
(b,d)
(a,b)
(a,b)
Founder Graph
5
3
6
4
2
1
8
7
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Founder Graph

• Includes m connected components, C1,Cm.
• Here C12, C21,3,5, C34,6,7,8
• The founder alleles assigned to different
  components appear in different genotyped
  individuals, by construction.
• Under random mating and Hardy-Weinberg
  equilibrium, the vectors of alleles assigned to
  different components are independent
• Each component can be processed individually.

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Singleton Components

• The vertices corresponding to alleles that never
  passed through genotyped individuals form
  singleton components.
• Any allele type can be assigned to singleton
  components.

Singleton component
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Singleton Components (cont.)
3
4
5
6
1
2
7
8
(a,b)
(a,b)
(a,c)
(b,d)
(a,b)
(a,b)
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Find compatible allelic assignments for
non-singleton components

• Identify the set of compatible alleles for each
  vertex. This is the intersection of the
  genotypes. attached to the edges incident to the
  vertex.

a,b n a,b a,b
a,b n b,d b
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Possible Allelic Assignments (example)
b
a
a,b
a,b
a,b
a,c
b,d
a,b,c,d
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Likelihood of descent graph. Sect 4
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Compatible Allelic Assignments

• Denote by A1,,Am the set of compatible allelic
  assignments obtained for each connected component
  at the end of the algorithm.
• Except for singleton components, each Ai contains
  0,1, or 2 assignments.
• If for some i, Ai is empty ? Prmv 0.
• The compatible assignments are those in the
  Cartesian product A1xxAm.

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Computing Prmv

• The probability of singleton components is 1 ? we
  can ignore them.
• Let ahi be an element of Ai (a vector of alleles
  assigned to the vertices of component Ci).

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Computing Prmv Complexity