Comput. Genomics, Lecture 5b Character Based Methods for Reconstructing Phylogenetic Trees: Maximum Parsimony

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Comput. Genomics, Lecture 5bCharacter Based
Methods for Reconstructing Phylogenetic
TreesMaximum Parsimony
Based on presentations by Dan Geiger, Shlomo
Moran, and Ido Wexler. Modified by Benny
Chor. References Durbin et al 7.4, Gusfield
17.1-17.3, SetubalMeidanis 6.1
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Phylogenetic Trees - Reminder

• Leaves represent objects (genes, species) being
  compared
• Internal nodes are hypothetical ancestral
  objects
• In a rooted tree, path from root to a node
  corresponds to a path in evolutionary time
• An unrooted tree specifies relationships among
  objects, but not evolutionary time

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Parsimony Based Approch

• Input Character data (aligned sequences)
• Goal/Output A labeled tree (labeled internal
• nodes) that explains the data with a
  minimal
• number of changes across edges

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Parsimony An Example

• Various trees that could explain the phylogeny
  of the following
• four sequences AAG, AAA, GGA, AGA. For
  example,

• Parsimony prefers the second tree to the first,
  because it requires less substitution events
  (three vs. four changes).

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Big and Small Parsimony

• Usually the approaches to finding a maximum
  parsimony
• tree have two separate components
• A search through the space of trees (BIG
  parsimony)
• Given a specific tree topology, find an
  assignment of ancestral labels to internal
  nodes as to the minimize the total number of
  changes across tree edges (small parsimony)

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Formally Big Parsimony

• Input Character data (aligned sequences)
• Goal/Output A labeled tree (labeled internal
• nodes) that minimizes number of changes
• across edges (over all trees and internal
  labelings).

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Formally Small Parsimony

• Input Character data (aligned sequences)
• and a tree with sequences at leaves.
• Goal/Output A labeling of internal nodes that
• minimizes number of changes across edges
• (over all internal labelings).

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Big, Small, and Weighted Parsimony

• Small parsimony has a linear time solution
• (Fitch algorithm).
• BIG parsimony is NP hard
• (easy reduction from vertex cover, VC).
• Weighted small parsimony also has a linear
  time solution
• (Sankoffs algorithm, dynamic programming).

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Small Parsimony Fitchs Algorithm

• Traverse tree up, from leaves to root,
  finding sets of possible ancestral states
  (labels) for each internal node.
• Traverse tree down, from root to leaves,
  determining ancestral states (labels) for
  internal nodes.
• Key observation Different sites are
  independent. Can solve one site at a time.

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Fitchs Algorithm Step 1

• Do a post-order (from leaves to root)
  traversal of tree
• Find out possible states Ri of internal node i
  with children j and k

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Fitchs Algorithm Step 1

• of changes union operations

T
T
AGT
CT
GT
C
T
G
T
A
T
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Fitchs Algorithm Step 2

• Do a pre-order (from root to leaves) traversal
  of tree
• Select state rj of internal node j with parent
  i

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Fitchs Algorithm Step 2
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Weighted Version

• Instead of assuming all state changes are unit
  cost
• ( ?equally likely), use different costs
  S(a,b) for
• different changes
• 1st step of algorithm is to propagate costs up
  through tree

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Weighted Version of Fitchs Algorithm

• Want to determine min. cost Ri(a)
• of assigning character a to node i
• for leaves

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Weighted Version of Fitchs Algorithm

• want to determine min. cost Ri(a)
• of assigning character a to node i
• for internal nodes

a
i
j
k
b
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Weighted Version of Fitchs Algorithm Step 2

• do a pre-order (from root to leaves) traversal
  of tree
• select minimal cost character for root
• For each internal node j, select character
  that produced minimal cost at parent i

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Big Parsimony Exploring the Space of Trees

• Weve considered small parsimony How to find
  the minimum number of changes for a given tree
  topology
• To solve big parsimony, need some search
  procedure for exploring the space of tree
  topologies
• There are
  unrooted trees on n leaves

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Exploring the Space of Trees

taxa (n) trees 4 15 5 105 6 945 8
135,135 10 30,405,375
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Does This Implies Big MP is Hard?

taxa (n) trees 4 15 5 105 6 945 8
135,135 10 30,405,375
Not necessarily There could be some smarter way
to zoom directly to best topology. But We
will show hardness of Big MP by a (simple)
reduction from vertex cover (VC).
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Big MP is NP Hard !

First, define VC and VC for triangle free
graphs. Then

• You will show a poly time reduction from VC to
  VC for triangle free graphs as part of home
  assignment (easy).
• In class, I will show a poly time reduction from
  
• VC for triangle free graphs to Big MP
• (old style, white board proof).
• This establishes NP hardness of Big MP.