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Phylogenetic Tree Construction
and Related Problems
Bioinformatics
Problems
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Multiple Sequence Alignment
Construction of Phylogenetic Trees
Determining Genomic Distance through
Rearrangements
Alignment of Multiple
Sequences
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We can extend the notion of alignment to
multiple strings:
-att-ct-ttactat-a-t
An alignment of strings S1…Sn is described
by strings S1’…Sn’
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Each Si’ contains the characters of Si in order
interspersed with spaces (-)
No position exists that contain spaces for all Si’
Scoring a Multiple Alignment
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Sum of pairs
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Consider each pair in the set of sequences,
determine the similarity score (using gap, match,
and mismatch weights) for each pair, and add all
pair-wise scores
Distance from consensus
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Consider all columns and count the total number
of differences from the consensus character
Variations: just count characters that differ from
consensus or have a difference score for each
differing character
Multiple Alignment Problem
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Formulation: Given sequences S1…Sn,
obtain an optimal multiple alignment of
the sequences
“Optimal” depends on multiple
alignment scoring method
No known (correct) efficient algorithms
for this problem
Strategies
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Brute force algorithm: consider all possible
alignments, then determine the one that
results in a best score (time complexity?)
Common Heuristic: Use regular (two-string)
alignment and then repeatedly add a string to
a growing alignment
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O(nm2) – where m is the the max string length
Does not always produce an optimal alignment
Phylogenetic Tree Construction
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Multiple alignments are often performed on
similar species
Next step: Construct an evolutionary tree of
these species
Input to problem: evolutionary distances
between each pair
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Not the same as similarity score
Example: edit distance--count number of indels
needed to transform one string to the other
Problem Formulation
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Given a set of species and evolutionary
distance between each pair (2d matrix
of numbers), construct a phylogenetic
tree consistent with the distances
Details needed:
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Characteristics/constraints of a tree
Notion of “consistent”
Phylogenetic Tree
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Rooted
Leaves are species
Internal nodes are speculated ancestors
Distances associated with each edge
Example
ancestor
1
4
ancestor
2
dog
bat
3
cat
Minimizing Distance Deviation
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Phylogenetic tree implies pairwise
distances (sum all edge distances in
path)
Compare with input distances to assess
consistency
Sum of Differences
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Alternative computations for deviation
(least squares)
Example
Suppose Input:
D(dog,cat): 4
D(dog,bat): 7
D(cat,bat): 10
ancestor
1
4
ancestor
2
dog
bat
3
cat
Implied by Tree:
D(dog,cat): 5
D(dog,bat): 7
D(cat,bat): 8
Difference = (5- 4) + (7-7) + (10-8) = 3
Character-based Tree
Construction
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Input:
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a set C of m characters
possible values for each character
a set S of n species where each element of S is associated
with a value for each character in C
Output: a phylogenetic tree T such that
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Elements in S are the leaves of T
Each internal node of T has an assigned value per character
Each character value induces a connected subtree of T
Mutations on Genomes
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Mutations can occur at the gene level
(indels of nucleotides) or at the genome
level (operations on genes)
At the genome level, for similar species,
the operations are rearrangements
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Reversals
Transpositions
Translocations
Genomes, Chromosomes, and
Permutations
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Genome: set of chromosomes
Chromosome: sequence of genes
Genes: unique across entire genome
Can simplify the representation of a
genome by arbitrarily concatenating the
chromosomes
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Result: a permutation of a set of unique
genes
Determining Genomic Distance
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Recall: Phylogenetic tree construction
need pair-wise distances
For similar species with the same set of
genes:
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Can use number of rearrangement
operations for distance
Common distance model: obtain the
fewest (optimal) rearrangements necessary
to transform genome X to genome Y
Some Assumptions Made
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Can limit allowed rearrangements to
some reasonable subset of possible
arrangements
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e.g., reversals apparently most common for
plant species
Sometimes, gene-blocks instead of
individual genes are the elements that
are permuted
Problem Formulations
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Given permutations (species) p and q of
the set {1…n}, find the shortest
sequence of rearrangements
(mutations) that transform p to q
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p.r1. r2 … rs = q
Given a permutation p, find the shortest
sequence of rearrangements that sort p
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p.r1. r2 … rs =
123…n