Bioinformatics Algorithms and Data Structures

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Bioinformatics Algorithms and Data Structures

• Chapter 17.1 Strings and Evolutionary Trees
• Lecturer Dr. Rose
• Slides by Dr. Rose
• April 8, 2003

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Strings and Evolutionary Trees

• the great Tree of Life fills with its dead and
  broken branches the crust of the earth, and
  covers the surface with is ever-branching and
  beautiful ramifications. - Darwin

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Strings and Evolutionary Trees

• There are three competing theories to creating
  classification trees
• Evolutionary taxonomy
• Numerical taxonomy
• Cladistics

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Strings and Evolutionary Trees

• Evolutionary taxonomy
• classification informed by evolutionary theory
• Fills in internal nodes corresponding to common
  ancestors.

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Strings and Evolutionary Trees

• Numerical taxonomy (Phenetics)
• Studies the relationship between groups of
  organisms based on the degree of similarity
• Similarity can be in terms of molecular,
  phenotypic or anatomical data.
• The resulting graph, which is a tree-like network
  is called a phenogram.
• Maximum Likelihood method.

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Strings and Evolutionary Trees

• Cladistics
• Characterized by character-state methods, e.g.,
  maximum parsimony.
• Guiding principle
• not all character states shared by organisms
  provide evolutionary information
• Important to restrict consideration to
  evolutionarily significant states.

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Strings and Evolutionary Trees

• Cladistics continued
• Willi Hennig Society
• http//www.cladistics.org/

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Strings and Evolutionary Trees

• Tree building algorithms
• Distance-based methods
• Input distance data such as sequence edit
  distance
• Output weighted tree with pairwise distances
  matching evolutionary distance
• We will consider data that is
• Ultrametric (section 17.1)
• Additive but not ultrametric (section 17.2)
• Nonadditive data (no section)

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Strings and Evolutionary Trees

• Tree building algorithms continued
• Maximum-parsimony methods
• Character-based methods
• Input character data (often aligned sequences)
• Output tree with
• input taxa at leaves
• Inferred taxa at internal nodes
• Goal minimize the total cost of mutations
• maximize parsimony.
• Seeks a tree that has the minimum cost over all
  possible trees

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Ultrametric trees and distances

• Before discussing ultrametric distances, consider
  additive distances
• Defn. Additive distances are distances which can
  be fitted to an unrooted tree such that all
  pairwise taxa distances are equal to the sum of
  the branch lengths connecting them. (Table and
  figure from http//imbs.massey.ac.nz/Research/MolE
  vol/Farside/DNA/00312.html)

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Ultrametric trees and distances

• Ultrametric distances are more constrained than
  additive distances.
• Defn. Ultrametric distances are distances that
• fit a tree so that the distance between any two
  taxa is equal to the sum of the branches joining
  them.
• for any three taxa i, j and k, the two largest
  distances are equal, i.e.,
• If dikgt djk then dik dij
• Else if dikgt dij then dik djk
• Else dij djk

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Ultrametric trees and distances

• Q What is an ultrametric tree?
• An ultrametric tree T for n-by-n symmetric
  distance matrix D has the following properties
• T has n leaves, one per unique row of D.
• Internal nodes are labeled by an entry from D and
  have two children.
• The numbers labeling internal nodes strictly
  decrease along any path from the root to a leaf.
• D(i, j) denotes the label of the least common
  ancestor of leaves i and j in T.
• The distances in D must be ultrametric.

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Ultrametric trees and distances

• Consider the following example from the textbook
• Verify the ultrametric condition, i.e., for any
  three taxa, two of the distances will be the same
  and larger than the third distance.

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Ultrametric trees and distances

• Interpretation of ultrametric as evolutionary
  trees
• The leaves are the existing OTUs
• The internal nodes are the divergence events
• A divergence event is a point where the
  evolutionary histories of two OTUs split.

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Ultrametric trees and distances

• Q If taxa A and B diverge at time t, which
  statements are implied by the meaning of
  divergence?
• A is the ancestor of B
• B is the ancestor of A
• Neither A nor B is an ancestor of the other.
• Neither A nor B have a living ancestor.

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Ultrametric trees and distances

• If the branching order time of each divergence
  is known
• The label at each internal node is the time of
  the divergent event corresponding to that node.
• The labels from the root to leaves must be
  strictly increasing.
• D(i, j) is the time that taxa i and j diverged.
• The author calls T a min-ultrametric tree.

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Ultrametric trees and distances

• Equivalently, if the evolutionary history is
  known
• The label at each internal node is the time that
  has passed since the divergent event
  corresponding to that node.
• The labels from the root to leaves must be
  strictly decreasing.
• D(i, j) is the time since taxa i and j diverged.
• T is an ultrametric tree for D.

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Ultrametric trees and distances

• Defn.The symmetric matrix D defines an
  ultrametric distance iff for any three indices i,
  j and k, the two largest distances are equal,
  i.e.,
• If dikgt djk then dik dij
• Else if dikgt dij then dik djk
• Else dij djk
• Call D ultrametric if it defines ultrametric
  distances.
• Thm. Distance matrix D has an ultrametric tree
  iff D is an ultrametric matrix. (Proof page 451)

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Ultrametric trees and distances

• Proof.
• (if T ultrametic then D ultrametric)
• If T is ultrametric (draw T)
• each internal node v is labeled D(i, j) where i
  and j are leaves and v is the least common
  ancestor.
• For any three leaves, i, j, k, in T, let u be
  the least common ancestor, then
• u is labeled by two of D(i, j), D(i, k), D(j,
  k), i.e., two of these are equal.
• Further more one of D(i, j), D(i, k), D(j, k)
  is smallest
• Therefore D is ultrametric.

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Ultrametric trees and distances

• Proof.
• ? (if D ultrametic then there is an ultrametric
  T)
• If D is ultrametric
• The number of distinct entries d in each row i
  defines the number of nodes from the root to leaf
  i.
• Each node in this path is labeled in decreasing
  order with a distinct label.
• Any node v on this path labeled D(i, j) must be
  the least common ancestor of leaves i and j.

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Ultrametric trees and distances

• Proof.? continued
• The path to leaf i partitions the n-1 remaining
  leaves in d-1 classes.
• Each distinct node on the path to i is labeled by
  the distance from i to to the leaves in that
  partition.
• Example

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Ultrametric trees and distances

• Proof.? continued
• We want to recursively find the ultrametric tree
  for each of the d-1 partitions and then combine
  them.
• Consider the partition defined by internal node
  v.
• Let j be a leaf contained in this partition.
• Let l be some other leaf. There are three cases
• l is in the same partition as j.
• l is in a partition between i and node v.
• l is in a partition between node v and the root.

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Ultrametric trees and distances

• Proof.? continued
• The three cases Let i A, j F
• l is in the same partition as j. example l B
• l is in a partition between i and v. example l
  D
• l is in a partition between v and the root.
  example l C

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Ultrametric trees and distances

• Proof.? continued
• Case 1 (i A, j F, l B)
• D(i, j) D(i, l) thus D(j, l) ? D(i, j). Why?
• ? So we can add the subtree containing j l.
  knowing that D(j, l) is correct.

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Ultrametric trees and distances
Typos in text

• Proof.? continued
• Case 2 (i A, j F, l D)
• D(i, l) lt D(i, j) thus D(i, j) D(j, l)
• ? So we can add the subtree at v containing j
  knowing that D(j, l) is correct, i.e. v is
  labeled correctly.

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Ultrametric trees and distances
Typos in text

• Proof.? continued
• Case 3 (i A, j F, l C)
• D(i, l) gt D(i, j) thus D(i, l) D(j, l)
• ? So we can add the subtree at v containing j
  knowing that D(j, l) is correct, i.e., it labels
  their least common ancestor .

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Ultrametric trees and distances

• Proof.? continued
• In each of the three cases, the ultrametric tree
  defined by v can be correctly attached to v.
• Hence, using recursion, we can construct the
  ultrametric tree T for D.

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Ultrametric trees and distances

• Gusfield presents two related theorems
• Thm. If D is ultrametric, then the ultrametric
  tree for D is unique.
• This is a consequence of the fact that the nodes
  that appear on path to a given node i must appear
  in every ultrametric tree for D.
• Thm. If D is ultrametric, then the ultrametric
  tree for D can be constructed in O(n2) time.

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Ultrametric trees and distances

• Given ultrametic data we can
• reconstruct evolutionary history.
• Find the relative divergence times
• Find the exact tree topology
• Q How do we get ultrametric data?
• Consider the molecular clock theory.

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Molecular Clock Theory

• Proposed by Emile Zucker and Linus Pauling.
• Idea accepted mutations occur at a constant rate
  for a given protein.
• There are three important issues
• Accepted mutations are those that still allow
  the protein to function properly.
• Lethal mutations will be selected against and
  should not accumulate.

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Molecular Clock Theory

• The theoretical clock rate is protein specific.
• Different proteins will have different clocks.
• Gusfield mentions hemoglobin and cytochrome c.
• Both are stable and similar in all mammals.
• However, hemoglobin mutates faster than
  cytochrome c.

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Molecular Clock Theory

• The implication is that the number of mutations
  is proportional to length of the time interval.
• Requirement the interval must be long enough.
• The length of an interval can be measured by the
  number of mutations.
• This requires that the clock be calibrated.

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Molecular Clock Theory

• Assumptions
• all DNA mutates at the same rate
• Observed accepted rate differences are due to
  different constraints
• Natural selection at the organism level
• Physical chemistry at the molecular level

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Molecular Clock Theory

• Q How would we use the molecular clock to
  collect ultrametric data?
• Find a common protein for two taxa of interest
• Determine the number of accepted mutational
  differences, say k.
• By molecular clock theory each taxa contributed
  k/2 accepted mutations.

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Molecular Clock Theory

• Do this for each pair of n taxa.
• Result n choose 2 numbers satisfying the
  requirement for an ultrametric tree.
• The ultrametric tree will describe the true
  evolutionary history for the n taxa.
• Great huh?

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Molecular Clock Theory

• If only the molecular clock theory was correct ?
• In the real world, the situation is complicated.
• Molecular clock rates can and do diverge.
• Sometimes there are common mutation rates.
• Sometimes the mutation rates diverge.

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Additive Distance Trees