Recombination and Pedigrees

1
Recombination and Pedigrees
Genealogies and Recombination The
ARG Recombination Parsimony The ARG and
Data Pedigrees Models and Data Pedigrees
ARGs Challenges Empirical Investigations Open
Questions
2
Recombination Histories Global Pedigrees
Acknowledgements Yun Song - Rune
Lyngsø - Mike Steel
3
Hudson Kaplans RM
0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1
1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0
1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
1 1 0 1
If you equate RM with expected number of
recombinations, this could be used as an
estimator. Unfortunately, RM is a gross
underestimate of the real number of
recombinations.
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Local Inference of Recombinations
5
Finding Minimal Recombination Histories
64 Bodmer Edwards Parsimony defined as
reconstruction principle 85 Hudson Kaplan uses
minimal recombination histories as observed
recombinations

• Attempts to find minimal histories of sequences
• Definition of recombination as Subtree Prune
  Regraft operations

1. J.J.Hein Reconstructing the history of
   sequences subject to Gene Conversion and
   Recombination. Mathematical Biosciences. (1990)
   98.185-200.
2. J.J.Hein A Heuristic Method to Reconstruct the
   History of Sequences Subject to Recombination.
   J.Mol.Evol. 20.402-411. 1993
3. Hein,J.J., T.Jiang, L.Wang K.Zhang (1996) "On
   the complexity of comparing evolutionary trees"
   Discrete Applied Mathematics 71.153-169
4. Song, Y.S. (2003) On the combinatorics of rooted
   binary phylogenetic trees. Annals of
   Combinatorics, 7365379
5. Song, Y.S. Hein, J. (2005) Constructing
   Minimal Ancestral Recombination Graphs. J. Comp.
   Biol., 12147169
6. Song, Y.S. Hein, J. (2004) On the minimum
   number of recombination events in the
   evolutionary history of DNA sequences. J. Math.
   Biol., 48160186.
7. Song, Y.S. Hein, J. (2003) Parsimonious
   reconstruction of sequence evolution and
   haplotype blocks finding the minimum number of
   recombination events, Lecture Notes in
   Bioinformatics, Proceedings of WABI'03,
   2812287302.
8. Lyngsø, Song and Hein (2005) Minimal
   Recombination Histories by Branch and Bound WABI

6
Minimal Number of Recombinations
The Kreitman data (1983) 11 sequences, 3200bp,
43(28) recoded, 9 different
Bi-partitions
How many neighbors?
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Two Adjacent Columns
1. RecDistT1,T2 is hard for large leaf number,
but can be automatically calculated by adding
DiamRec trivial columns and only considering 1
recombination neighbors.
2. Infinite Site Assumption Local Trees must
contain Local Bipartition
8
Metrics on Trees based on subtree transfers.
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Tree Combinatorics and Neighborhoods
Observe that the size of the unit-neighbourhood
of a tree does not grow nearly as fast as the
number of trees
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1
4
2
3
5
6
7
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The Minimal Recombination History for the
Kreitman Data
Methods of rec events obtained
Hudson Kaplan (1985) 5
Myers Griffiths (2003) 6
Song Hein (2004). Set theory based approach. 7
Song Hein (2003). Tree scanning using DP Lyngsø, Song Hein (2006). Massive Acceleration using Branch and Bound Algorithm. Lyngsø, Song Hein (2006). Minimal number of Gene Conversions (in prep.) 7 7 5-2/6-1
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- recombination 27 ACs
recombination 3108 ACs
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Ancestral configurations to 2 sequences with 2
segregating sites
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Counting Recursion
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Counting Branch and Bound Algorithm
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Time versus Spatial Coalescent-Recombination
(Griffiths, 1981 Hudson, 1983 - Wiuf Hein,
1999)
Temporal Process
Spatial Process
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Time versus Spatial 2 Pedigrees
Elston-Stewart (1971) -Temporal Peeling
Algorithm
Father
Mother
Condition on parental states Recombination and
mutation are Markovian
Lander-Green (1987) - Genotype Scanning
Algorithm
Father
Mother
Condition on paternal/maternal inheritance Recombi
nation and mutation are Markovian
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Time versus Spatial 3 Phylogenetic Alignment

• Optimisation Algorithms
• indels of length 1 (David Sankoff, 1973)
  Spatial
• indels of length k (Bjarne Knudsen, 2003)
  Temporal
• Statistical Alignment

Temporal
Spatial
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minARGs Recombination Events Local Trees
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Likelihood Calculations on the e-ARG
010 010 101 101 110
Example
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Reconstructing global pedigrees
Superpedigrees Steel and Hein, 2005
The gender-labeled pedigrees for all pairs,
defines global pedigree
Gender-unlabeled pedigrees doesnt!!
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Reconstructing global pedigrees Links and lassos
Steel and Hein, 2005
Gender-labeled links and lassos determine the
global pedigree.
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Benevolent Mutation and Recombination Process

• All embedded phylogenies are observable
• Do they determine the pedigree?

Counter example
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Summary and Future

• Minimal Recombination Histories
• Likelihood Calculations
• Global Pedigrees Inferring Pedigrees from
  Genomes

Recombination Remove infinite site assumption
Investigate MCMC algorithms Pedigrees Data
Analysis Algorithms
Presentation can be found at
http//mathgen.stats.ox.ac.uk/bioinformatics/
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To Do

• Hudson Slide
• Neighbor trees
• Literature and History