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• Gene: A sequence of nucleotides coding for protein
• Gene Prediction Problem: Determine the beginning and end positions of genes in a genome
Gene Prediction: Computational Challenge
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Central Dogma and Splicingexon1 exon2 exon3
intron1 intron2
transcription
translation
splicing
exon = codingintron = non-coding
Batzoglou
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Splice site detection
5’ 3’Donor site
Position
% -8 … -2 -1 0 1 2 … 17
A 26 … 60 9 0 1 54 … 21C 26 … 15 5 0 1 2 … 27G 25 … 12 78 99 0 41 … 27T 23 … 13 8 1 98 3 … 25
From lectures by Serafim Batzoglou (Stanford)
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Six Frames in a DNA Sequence
• stop codons – TAA, TAG, TGA• start codons - ATG
GACGTCTGCTTTGGAGAACTACATCAACCGGACTGTGGCTGTTATTACTTCTGATGGCAGAATGATTGTG
CTGCAGACGAAACCTCTTGATGTAGTTGGCCTGACACCGACAATAATGAAGACTACCGTCTTACTAACAC
GACGTCTGCTTTGGAGAACTACATCAACCGGACTGTGGCTGTTATTACTTCTGATGGCAGAATGATTGTGGACGTCTGCTTTGGAGAACTACATCAACCGGACTGTGGCTGTTATTACTTCTGATGGCAGAATGATTGTGGACGTCTGCTTTGGAGAACTACATCAACCGGACTGTGGCTGTTATTACTTCTGATGGCAGAATGATTGTG
CTGCAGACGAAACCTCTTGATGTAGTTGGCCTGACACCGACAATAATGAAGACTACCGTCTTACTAACACCTGCAGACGAAACCTCTTGATGTAGTTGGCCTGACACCGACAATAATGAAGACTACCGTCTTACTAACACCTGCAGACGAAACCTCTTGATGTAGTTGGCCTGACACCGACAATAATGAAGACTACCGTCTTACTAACAC
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Codon Usage in Human Genome
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Promoter Structure in Prokaryotes (E.Coli)
Transcription starts at offset 0.
• Pribnow Box (-10)
• Gilbert Box (-30)
• Ribosomal Binding Site (+10)
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TestCode Statistics
• Define a window size no less than 200 bp, slide the window the sequence down 3 bases. In each window:
• Calculate for each base {A, T, G, C}
• max (n3k+1, n3k+2, n3k) / min ( n3k+1, n3k+2, n3k)
• Use these values to obtain a probability from a lookup table (which was a previously defined and determined experimentally with known coding and noncoding sequences
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Distribution of Each Base
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Position Parameter
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TestCode Sample Output
Coding
No opinion
Non-coding
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Popular Gene Prediction Algorithms
• GENSCAN: uses Hidden Markov Models (HMMs)
• TWINSCAN
• Uses both HMM and similarity (e.g., between human and mouse genomes)
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Statistics
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Weight
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TestCode Method
• Compute A,C,G,T position and content parameters
• Look up from probability of coding value and get p1, p2, …p8
• Get corresponding weights w1, w2, …w8
• Compute p1 w1+ p2 w2 +…+p8 w8
• This is the indicator of coding function
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Comparing Genes in Two Genomes
• Small islands of similarity corresponding to similarities between exons
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Finding Local AlignmentsUse local alignments to find all islands of similarity
Human Genome
Frog G
enes (know
n)
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Exon Chaining Problem: Graph Representation
• This problem can be solved with dynamic programming in O(n) time.
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Infeasible Chains Red local similarities form two non -overlapping
intervals but do not form a valid global alignment
Human Genome
Frog G
enes (know
n)
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Genome Rearrangements
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Simple Rearrangements
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Phylogenetic Reconstruction
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Rearrangement Phylogeny
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Turnip vs Cabbage: Look and Taste Different• Although cabbages and turnips share a
recent common ancestor, they look and taste different
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Turnip vs Cabbage: Comparing Gene Sequences Yields No Evolutionary Information
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Turnip vs Cabbage: Almost Identical mtDNA gene sequences
• In 1980s Jeffrey Palmer studied evolution of plant organelles by comparing mitochondrial genomes of the cabbage and turnip
• 99% similarity between genes• These surprisingly identical gene sequences
differed in gene order• This study helped pave the way to analyzing
genome rearrangements in molecular evolution
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Turnip vs Cabbage: Different mtDNA Gene Order
• Gene order comparison:
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Turnip vs Cabbage: Different mtDNA Gene Order
• Gene order comparison:
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Turnip vs Cabbage: Different mtDNA Gene Order
• Gene order comparison:
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Turnip vs Cabbage: Different mtDNA Gene Order
• Gene order comparison:
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Turnip vs Cabbage: Different mtDNA Gene Order• Gene order comparison:
Before
After
Evolution is manifested as the divergence in gene order
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Transforming Cabbage into Turnip
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•What are the similarity blocks and how to find them?
•What is the architecture of the ancestral genome?
•What is the evolutionary scenario for transforming one genome into the other?
Unknown ancestor~ 75 million years ago
Mouse (X chrom.)
Human (X chrom.)
Genome rearrangements
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History of Chromosome X
Rat Consortium, Nature, 2004
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Reversals
• Blocks represent conserved genes.
1 32
4
10
56
8
9
7
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
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Reversals1 32
4
10
56
8
9
7
1, 2, 3, -8, -7, -6, -5, -4, 9, 10
Blocks represent conserved genes. In the course of evolution or in a clinical context, blocks
1,…,10 could be misread as 1, 2, 3, -8, -7, -6, -5, -4, 9, 10.
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Reversals and Breakpoints1 32
4
10
56
8
9
7
1, 2, 3, -8, -7, -6, -5, -4, 9, 10
The reversion introduced two breakpoints(disruptions in order).
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Reversals: Example
5’ ATGCCTGTACTA 3’
3’ TACGGACATGAT 5’
5’ ATGTACAGGCTA 3’
3’ TACATGTCCGAT 5’
Break and Invert
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Types of Rearrangements
Reversal1 2 3 4 5 6 1 2 -5 -4 -3 6
Translocation1 2 3 44 5 6
1 2 6 4 5 3
1 2 3 4 5 6
1 2 3 4 5 6
Fusion
Fission
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Comparative Genomic Architectures: Mouse vs Human Genome
• Humans and mice have similar genomes, but their genes are ordered differently
• ~245 rearrangements• Reversals• Fusions• Fissions• Translocation
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Waardenburg’s Syndrome: Mouse Provides Insight into Human Genetic Disorder
• Waardenburg’s syndrome is characterized by pigmentary dysphasia
• Gene implicated in the disease was linked to human chromosome 2 but it was not clear where exactly it is located on chromosome 2
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Waardenburg’s syndrome and splotch mice
• A breed of mice (with splotch gene) had similar symptoms caused by the same type of gene as in humans
• Scientists succeeded in identifying location of gene responsible for disorder in mice
• Finding the gene in mice gives clues to where the same gene is located in humans
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Comparative Genomic Architecture of Human and Mouse Genomes
To locate where corresponding gene is in humans, we have to analyze the relative architecture of human and mouse genomes
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Reversals: Example
= 1 2 3 4 5 6 7 8
(3,5)
1 2 5 4 3 6 7 8
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Reversals: Example
= 1 2 3 4 5 6 7 8
(3,5)
1 2 5 4 3 6 7 8
(5,6)
1 2 5 4 6 3 7 8
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Reversals and Gene Orders• Gene order is represented by a permutation 1------ i-1 i i+1 ------j-1 j j+1 -----
n
1------ i-1 j j-1 ------i+1 i j+1 -----n
Reversal ( i, j ) reverses (flips) the elements from i to j in
,j)
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Reversal Distance Problem
• Goal: Given two permutations, find the shortest series of reversals that transforms one into another
• Input: Permutations and
• Output: A series of reversals 1,…t transforming into such that t is minimum
• t - reversal distance between and • d(, ) - smallest possible value of t, given and
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Sorting By Reversals Problem
• Goal: Given a permutation, find a shortest series of reversals that transforms it into the identity permutation (1 2 … n )
• Input: Permutation
• Output: A series of reversals 1, … t transforming into the identity permutation such that t is minimum
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Sorting By Reversals: Example
• t =d( ) - reversal distance of • Example : = 3 4 2 1 5 6 7 10 9 8 4 3 2 1 5 6 7 10 9 8 4 3 2 1 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 So d( ) = 3
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Sorting by reversals: 5 stepsStep 0: 2 -4 -3 5 -8 -7 -6 1Step 1: 2 3 4 5 -8 -7 -6 1Step 2: 2 3 4 5 6 7 8 1Step 3: 2 3 4 5 6 7 8 -1Step 4: -8 -7 -6 -5 -4 -3 -2 -1Step 5: g 1 2 3 4 5 6 7 8
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Sorting by reversals: 4 stepsStep 0: 2 -4 -3 5 -8 -7 -6 1Step 1: 2 3 4 5 -8 -7 -6 1Step 2: -5 -4 -3 -2 -8 -7 -6 1Step 3: -5 -4 -3 -2 -1 6 7 8Step 4: g 1 2 3 4 5 6 7 8
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Sorting by reversals: 4 stepsStep 0: 2 -4 -3 5 -8 -7 -6 1Step 1: 2 3 4 5 -8 -7 -6 1Step 2: -5 -4 -3 -2 -8 -7 -6 1Step 3: -5 -4 -3 -2 -1 6 7 8Step 4: g 1 2 3 4 5 6 7 8
What is the reversal distance for this permutation? Can it be sorted in 3 steps?
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= 23…n-1n
• A pair of elements i and i + 1 are adjacent if
i+1 = i + 1• For example: = 1 9 3 4 7 8 2 6 5• (3, 4) or (7, 8) and (6,5) are adjacent pairs
Adjacencies and Breakpoints
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There is a breakpoint between any adjacent element that are non-consecutive:
= 1 9 3 4 7 8 2 6 5
• Pairs (1,9), (9,3), (4,7), (8,2) and (2,5) form breakpoints of permutation
• b() - # breakpoints in permutation
Breakpoints: An Example
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Adjacency & Breakpoints•An adjacency - a pair of adjacent elements that are consecutive
• A breakpoint - a pair of adjacent elements that are not consecutive
π = 5 6 2 1 3 4
0 5 6 2 1 3 4 7adjacencies
breakpoints
Extend π with π0 = 0 and π7 = 7
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• We put two elements 0 =0 and n + 1=n+1 at the ends of Example:
Extending with 0 and 10
Note: A new breakpoint was created after extending
Extending Permutations
= 1 9 3 4 7 8 2 6 5
= 0 1 9 3 4 7 8 2 6 5 10
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Each reversal eliminates at most 2 breakpoints.
= 2 3 1 4 6 50 2 3 1 4 6 5 7 b() = 50 1 3 2 4 6 5 7 b() = 40 1 2 3 4 6 5 7 b() = 20 1 2 3 4 5 6 7 b() = 0
Reversal Distance and Breakpoints
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Each reversal eliminates at most 2 breakpoints. This implies: reversal distance ≥ #breakpoints / 2
= 2 3 1 4 6 50 2 3 1 4 6 5 7 b() = 50 1 3 2 4 6 5 7 b() = 40 1 2 3 4 6 5 7 b() = 20 1 2 3 4 5 6 7 b() = 0
Reversal Distance and Breakpoints
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Breakpoint Graph1) Represent the elements of the permutation π = 2 3 1 4 6 5 as
vertices in a graph (ordered along a line)
0 2 3 1 4 6 5 7
2) Connect vertices in order given by π with black edges (black path)
3) Connect vertices in order given by 1 2 3 4 5 6 with grey edges (grey path)
4) Superimpose black and grey paths
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Two Equivalent Representations of the Breakpoint Graph
0 2 3 1 4 6 5 7
0 1 2 3 4 5 6 7
• Consider the following Breakpoint Graph
• If we line up the gray path (instead of black path) on a horizontal line, then we would get the following graph
• Although they may look different, these two graphs are the same
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What is the Effect of the Reversal ?
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 7
• The gray paths stayed the same for both graphs• There is a change in the graph at this point• There is another change at this point
How does a reversal change the breakpoint graph?
Before: 0 2 3 1 4 6 5 7
After: 0 2 3 5 6 4 1 7
• The black edges are unaffected by the reversal so they remain the same for both graphs
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A reversal affects 4 edges in the breakpoint graph
0 1 2 3 4 5 6 7
• A reversal removes 2 edges (red) and replaces them with 2 new edges (blue)
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Effects of Reversals (Continued)Case 2:
Both edges belong to different cycles
• Remove the center black edges and replace them with new black edges
• After the replacement, there now exists 1 cycle instead of 2 cycles
c(πρ) – c(π) = -1
Therefore, for every permutation π and reversal ρ, c(πρ) – c(π) ≤ 1
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Reversal Distance and Maximum Cycle Decomposition
• Since the identity permutation of size n contains the maximum cycle decomposition of n+1, c(identity) = n+1
• c(identity) – c(π) equals the number of cycles that need to be “added” to c(π) while transforming π into the identity
• Based on the previous theorem, at best after each reversal, the cycle decomposition could increased by one, then:
d(π) = c(identity) – c(π) = n+1 – c(π)
• Yet, not every reversal can increase the cycle decomposition
Therefore, d(π) ≥ n+1 – c(π)
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Signed Permutations• Up to this point, all permutations to sort were
unsigned• But genes have directions… so we should
consider signed permutations5’ 3’
= 1 -2 - 3 4 -5
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Signed Permutation• Genes are directed fragments of DNA and we represent a genome by a signed permutation
• If genes are in the same position but there orientations are different, they do not have the equivalent gene order
• For example, these two permutations have the same order, but each gene’s orientation is the reverse; therefore, they are not equivalent gene sequences
1 2 3 4 5
-1 2 -3 -4 -5
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From Signed to Unsigned Permutation
0 +3 -5 +8 -6 +4 -7 +9 +2 +1 +10 -11 12
• Begin by constructing a normal signed breakpoint graph
• Redefine each vertex x with the following rules:
If vertex x is positive, replace vertex x with vertex 2x-1 and vertex 2x in that order
If vertex x is negative, replace vertex x with vertex 2x and vertex 2x-1 in that order
The extension vertices x = 0 and x = n+1 are kept as it was before
0 3a 3b 5a 5b 8a 8b 6a 6b 4a 4b 7a 7b 9a 9b 2a 2b 1a 1b 10a 10b 11a 11b 23
0 5 6 10 9 15 16 12 11 7 8 14 13 17 18 3 4 1 2 19 20 22 21 23
+3 -5 +8 -6 +4 -7 +9 +2 +1 +10 -11
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From Signed to Unsigned Permutation (Continued)
0 5 6 10 9 15 16 12 11 7 8 14 13 17 18 3 4 1 2 19 20 22 21 23
• Construct the breakpoint graph as usual
• Notice the alternating cycles in the graph between every other vertex pair
• Since these cycles came from the same signed vertex, we will not be performing any reversal on both pairs at the same time; therefore, these cycles can be removed from the graph
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Interleaving Edges
0 5 6 10 9 15 16 12 11 7 8 14 13 17 18 3 4 1 2 19 20 22 21 23
• Interleaving edges are grey edges that cross each other
These 2 grey edges interleave
Example: Edges (0,1) and (18, 19) are interleaving
• Cycles are interleaving if they have an interleaving edge
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Interleaving Graphs
0 5 6 10 9 15 16 12 11 7 8 14 13 17 18 3 4 1 2 19 20 22 21 23
• An Interleaving Graph is defined on the set of cycles in the Breakpoint graph and are connected by edges where cycles are interleaved
A
BC
E
F
0 5 6 10 9 15 16 12 11 7 8 14 13 17 18 3 4 1 2 19 20 22 21 23
A
BC
E
F
D
D AB
C
E F
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Interleaving Graphs (Continued)
AB
C
D E F
• Oriented cycles are cycles that have the following form
F
C
• Unoriented cycles are cycles that have the following form
• Mark them on the interleave graph
E
• In our example, A, B, D, E are unoriented cycles while C, F are oriented cycles
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Hurdles• Remove the oriented components from the interleaving graph
AB
C
D E F
• The following is the breakpoint graph with these oriented components removed
• Hurdles are connected components that do not contain any other connected components within it
A
B D
E
Hurdle
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Reversal Distance with Hurdles
• Hurdles are obstacles in the genome rearrangement problem
• They cause a higher number of required reversals for a permutation to transform into the identity permutation
• Taking into account of hurdles, the following formula gives a tighter bound on reversal distance:
d(π) ≥ n+1 – c(π) + h(π)
• Let h(π) be the number of hurdles in permutation π
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Median Problem
Goal: find M so that DAM+DBM+DCM is minimized
NP hard for most metric distances
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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
$
1
-1
3
-3
2
-3
2 3 $
-3
.
$3
-3
...
...
...
...
.
.
.
.
.
.
.
.
.
.
.
...$-3
3 $
‹ 1, 2, 3 ›
‹ 1, 2, -3 ›
‹ 1, 2 › ‹ 3 ›
‹ 1, 2 › ‹ -3 ›
Genome Enumeration
For genomes on gene {1,2,3}
2
-2
2
-2
2
-2
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Genome Enumeration for Multichromosome Genomes
$
1
-1
3
-3
2
-3
2 3 $
-3
.
$3
-3
...
...
...
...
.
.
.
.
.
.
.
.
.
.
.
...$-3
3 $
‹ 1, 2, 3 ›
‹ 1, 2, -3 ›
‹ 1, 2 › ‹ 3 ›
‹ 1, 2 › ‹ -3 ›
Genome Enumeration
For genomes on gene {1,2,3}
• Red line: end fixing• Black line: edge fixing
2
-2
2
-2
2
-2
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Rearrangement Phylogeny
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Compute A Given Tree (Start)
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Compute A Given Tree (First Median)
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Compute A Given Tree (Second Median)
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Compute A Given Tree (Third Median)
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Compute A Given Tree (After 1st Iteration)
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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 ExampleA acgcaa
B acataa
C atgtca
D gcgtta
A B C D A C B D A D C B
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All Possible Evolutionary Paths (Column 1)
a a a g
a c g t a c g t
a c g t
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Likelihood for One Path
a a a g
a g
t
)()()()()()(1 ggPagPgtPaaPaaPatPL
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Sum of All Paths (Column 1)
a a a g
a c g t a c g t
a c g t
64
11column
iiLP
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Whole Sequence
A B C D
5
1icolumn 1 Tree
i
PP
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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 Results (Equal Content)
80% inversion, 20% transposition
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Experimental Results (Unequal)
90% inversion, 10% of del/ins/dup, 5-30 genes per segment
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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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98
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 Example—New Genomes1 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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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