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Combinatorial and Statistical Approaches in Gene Rearrangement Analysis

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Title: Combinatorial and Statistical Approaches in Gene Rearrangement Analysis


1
Combinatorial and Statistical Approaches in Gene
Rearrangement Analysis
  • Jijun Tang
  • Computer Science and Engineering
  • University of South Carolina
  • jtang_at_cse.sc.edu
  • (803) 777-8923

2
Outline
  • Backgrounds
  • Branch-and-Bound Algorithms for the Median
    Problem
  • Maximum Likelihood Methods for Phylogenetic
    Reconstruction
  • Post-Analysis
  • Conclusions

3
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4
Simple Rearrangements
5
Phylogenetic Reconstruction
6
Rearrangement Phylogeny
7
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8
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9
Median Problem
Goal find M so that DAMDBMDCM is minimized NP
hard for most metric distances
10
Multichromosomal Reversal Median problem
  • To find a median genome that minimizes the
    summation of the multichromosomal HP distances on
    the three edges
  • Events considered reversal, translocation,
    fusion, fission
  • Exact and heuristic solvers exist for the
    Unichromosomal Reversal Median Problem (reversals
    are the only events)

11
Capless Breakpoint Graph
  • Genome A ? Non-perfect Matching M(A)
  • Let a,b be adjacency genes in A. Then (at,bh) is
    an edge in M(A)
  • A genome is composed of a set of edges and ends.
  • Matchings naturally correspond to Undirected
    Genomes (Flipping of chromosomes does not alter
    matchings)

12
Example
  • Example Genomes
  • A -5, 1, 6, 3 , 2, 4
  • B 1, 6 , -5, -4, -3, -2

Adjacency Graph
13
Capless Breakpoint Graph
B-end
A-end
  • Denote C(A,B) Cycles, AB AB-Paths, AA
    AA-paths, BB BB-paths in G(A,B), n
    genes
  • n 6,C(A,B) 1,AB 4,
  • dHP 6-1-4/2 3

14
A Lower Bound of the HP Distance
  • A simpler lower bound only contains genes,
    cycles, paths.
  • Derived from Hannenhalli, Pevzner 1995
  • dHP (A,B)n C(A,B) - AB/2 AA - BB
  • Pseudo-cycle of A and B

15
Pseudo-cycle distance Median Problem
  • Pseudo-cycle distance
  • Pseudo-cycle distance Median Problem (PMP) to
    find a median genome that minimizes the summation
    of the Pseudo-cycle distance on the three edges
  • We use the Pseudo-cycle distance as a lower bound
    for the HP distance to derive a RMP solver

16
Branch-and-Bound Algorithm
  • Enumerate the solution genomes gene by gene
    (Genome Enumeration)
  • After enumerated a gene, compute an upper bound
    based on the partial solution genome
  • Bound check whether the upper bound of the
    partial solution is less than a criteria
  • Branch
  • If it is true, the partial genome is discarded,
    enumerate another gene
  • Otherwise update the criteria and continue
    enumeration

17
Genome Enumeration for Multichromosome Genomes
Genome Enumeration For genomes on gene 1,2,3
2
2
2
-2
-2
-2
18
Features
  • Main Components
  • Contraction Operation
  • Upper Bound on the number of pseudo-cycles
  • Genome enumeration
  • Extension of Capraras method for unichromosomal
    genomes (1999)

19
Contraction Operation
  • Contraction eat,bh on M(A) M(A)/e

20
Upper Bound on the Number of Pseudo-cycles
  • Let S be a genome and ZG1, G2, G3 a set of
    three input genomes
  • The maximal ?(S,Z) is denoted by ?
  • Based on triangle inequality, an upper bound on
    the number of pseudo-cycles can be derived

21
Notes
  • qn- ? is the lower bound of the sum of
    pseudo-cycle distances between any S and each
    genome in Z G1, G2, G3
  • Given an edge e, assume genome S contains e and
    maximizes ?(S,Z) let ZG1/e, G2/e, G3/e, and
    assume S maximizes Z?(S,Z), then S S?e

22
Upper Bound Test
  • In a step of the algorithm, the current partial
    solution is Sie1,e2,,ei
  • The upper bound of ?(S,Z) of genoms containing Si
    is the following
  • Let UB be the current upper bound
  • If UBSiltUB, then the best upper bound of the
    genomes containing Si is worse than UB

23
Branch-and-Bound Algorithm for Multichromosomal
Genomes
  • Compute an initial Upper Bound (UB) from the
    input genomes.
  • In each step, either an end or an edge is fixed
    in the solution.
  • End Fixing Mark a node as an end of a
    chromosome.
  • Edge Fixing Fix an edge e to the current partial
    solution genome Si.

24
Genome Enumeration for Multichromosome Genomes
Genome Enumeration For genomes on gene 1,2,3
2
2
2
-2
-2
-2
  • Red line end fixing
  • Black line edge fixing

25
Properties
  • Can be extended to compute a given tree using
    iterative or progressive approaches
  • However, median computation is still difficult
  • Large nuclear genomes
  • Complex events
  • We also need to search the best tree from the
    large tree space
  • N species
  • 20 species

26
Statistical Approaches
  • Combinatorial approaches are the focus of genome
    rearrangement research
  • Only one MCMC method exists
  • Maximum Likelihood methods have been very popular
    in sequence phylogenetic analysis
  • Bootstrapping (data resampling) is a popular
    method to assess quality of obtained trees
  • Hard to directly apply ML and bootstrapping to
    gene order

27
Sequence ML Phylogeny
  • For each position, generate all possible tree
    structures
  • Based on the evolutionary model, calculate
    likelihood of these trees and sum them to get the
    column likelihood
  • Calculate tree likelihood by multiplying the
    likelihood for each position
  • Choose tree with the greatest likelihood

28
Example
A acgcaa
B acataa
C atgtca
D gcgtta
29
All Possible Evolutionary Paths (Column 1)
a c g t
a c g t
a c g t
30
Likelihood for One Path
a
a
a
g
31
Sum of All Paths (Column 1)
a c g t
a c g t
a c g t
32
Whole Sequence
33
MLBE
  • Convert the gene-orders into binary sequences
    based on adjacencies
  • Convert the binary sequences into protein or DNA
    sequence
  • Use RAxML to compute a ML tree on the sequences
  • Binary encoding was used before for parsimony
    analysis, with reasonable results

34
Binary Encoding
35
MLBE Sequences
36
Experimental Setup
  • Generate random trees of N taxa
  • Each tree is equally likely
  • Birth-death model is preferred
  • Starting from the root, apply r events along each
    edge
  • r is the expected number of events
  • Actual number is a sample between 12r
  • Comparing the inferred tree with the true tree
    using RF rate

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Experimental Results (Equal Content 1)
80 inversion, 20 transposition
39
Experimental Results (Equal Content 2)
80 inversion, 20 transposition
40
Experimental Results (Unequal 1)
90 inversion, 10 of del/ins/dup, 5-30 genes per
segment
41
Experimental Results (Unequal 2)
90 inversion, 10 of del/ins/dup, 5-30 genes per
segment
42
Multistate Endocing
43
MLME Results (200 genes 20 genomes)
44
MLME Results (1000 genes 20 genomes)
45
Post Analysis
  • Bootstrapping has been widely used to assess the
    quality of sequence phylogeny
  • The same procedure is impossible for gene order
    data since there is only one character
  • We tested the procedure of jackknifing through
    simulated data to obtain
  • Is jackknifing useful
  • The best jackknifing rate
  • What is the threshold of the support values

46
DNA bootstrapping
47
Bootstrapping Results
48
Jackknifing Procedure
  • Generate a new dataset by removing half of the
    genes from the original genomes (orders are
    preserved)
  • Compute a tree on the new dataset
  • Repeat K times and obtain K replicates
  • Obtain a consensus tree with support values

49
An ExampleNew Genomes
  • 1 2 3 4 5 6 7 8 9 10
  • 1 -4 5 2 8 10 9 -7 -6 3

1 3 5 7 9 1 5 9 -7 3
50
Jackknifing Rate
51
Support Value Threshold - FP
Up to 90 FP can be identified with 85 as the
threshold
52
Trees with FP
53
Support Value Threshold - FN
54
Low Support Branches
55
Jackknife Properties
  • Jackknifing is necessary and useful for gene
    order phylogeny, and a large number of errors can
    be identified
  • 40 jackknifing rate is reasonable
  • 85 is a conservative threshold, 75 can also be
    used
  • Low support branches should be examined in detail

56
Conclusions
  • Great progress has been made in genome
    rearrangement research
  • We are able to handle real size data
  • Now the question is what data
  • Data quality and biological modeling
  • Ancestral genome reconstruction is still
    difficult
  • Putting everything together has just started

57
Thank You!
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