Combinatorial and Statistical Approaches in Gene Rearrangement Analysis

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Combinatorial and Statistical Approaches in Gene
Rearrangement Analysis

• Jijun Tang
• Computer Science and Engineering
• University of South Carolina
• jtang_at_cse.sc.edu
• [phone]

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Outline

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

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Simple Rearrangements
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Phylogenetic Reconstruction
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Rearrangement Phylogeny
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Median Problem
Goal find M so that DAMDBMDCM is minimized NP
hard for most metric distances
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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)

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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)

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Example

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

Adjacency Graph
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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

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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

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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

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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

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Genome Enumeration for Multichromosome Genomes
Genome Enumeration For genomes on gene 1,2,3
2
2
2
-2
-2
-2
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Features

• Main Components
• Contraction Operation
• Upper Bound on the number of pseudo-cycles
• Genome enumeration
• Extension of Capraras method for unichromosomal
  genomes (1999)

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Contraction Operation

• Contraction eat,bh on M(A) M(A)/e

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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

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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

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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

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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.

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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

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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

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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

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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

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Example
A acgcaa
B acataa
C atgtca
D gcgtta
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All Possible Evolutionary Paths (Column 1)
a c g t
a c g t
a c g t
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Likelihood for One Path
a
a
a
g
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Sum of All Paths (Column 1)
a c g t
a c g t
a c g t
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Whole Sequence
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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

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Binary Encoding
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MLBE Sequences
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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
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Experimental Results (Equal Content 2)
80 inversion, 20 transposition
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Experimental Results (Unequal 1)
90 inversion, 10 of del/ins/dup, 5-30 genes per
segment
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Experimental Results (Unequal 2)
90 inversion, 10 of del/ins/dup, 5-30 genes per
segment
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Multistate Endocing
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MLME Results (200 genes 20 genomes)
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MLME Results (1000 genes 20 genomes)
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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

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DNA bootstrapping
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Bootstrapping Results
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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

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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
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Jackknifing Rate
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Support Value Threshold - FP
Up to 90 FP can be identified with 85 as the
threshold
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Trees with FP
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Support Value Threshold - FN
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Low Support Branches
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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

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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

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Thank You!