Solving Phylogenetic Trees

1
Solving Phylogenetic Trees

• Benjamin Loyle
• March 16, 2004
• Cse 397 Intro to MBIO

2
Table of Contents

• Problem Term Definitions
• A DCM-NJ Solution
• Performance Measurements
• Possible Improvements

3
Phylogeny
From the Tree of the Life Website,University of
Arizona
Orangutan
Human
Gorilla
Chimpanzee
4
DNA Sequence Evolution
5
Problem Definition

• The Tree of Life
• Connecting all living organisms
• All encompassing
• Find evolution from simple beginnings
• Even smaller relations are tough
• Impossible
• Infer possible ancestral history.

6
So what.

• Genome sequencing provides entire map of a
  species, why link them?
• We can understand evolution
• Viable drug testing and design
• Predict the function of genes
• Influenza evolution

7
Why is that a problem?

• Over 8 million organisms
• Current solutions are NP-hard
• Computing a few hundred species takes years
• Error is a very large factor

8
What do we want?

• Input
• A collection of nodes such as taxa or protein
  strings to compare in a tree
• Output
• A topological link to compare those nodes to each
  other
• When do we want it?
• FAST!

9
Preparing the input

• Create a distance matrix
• Sum up all of the known distances into a matrix
  sized n x n
• N is the number of nodes or taxa
• Found with sequence comparison

10
Distance Matrix
Take 5 separate DNA strings A GATCCATGA B
GATCTATGC C GTCCCATTT D AATCCGATC E
TCTCGATAG The distance between A and B is 2 The
distance between A and C is 4 This is subjective
based on what your criteria are.
11
Distance Matrix

• Lets start with an example matrix

A
B
C
D
E
A
B
C
D
E
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Lets make it simple (constrain the input)

• Lets keep the distance between nodes within a
  certain limit
• From F -gt G
• F and G have the largest distance they are the
  most dissimilar of any nodes.
• This is called the diameter of the tree
• Lets keep the length of the input (length of the
  strings) polynomial.

13
ERROR?!?!!?

• All trees are inferred, how do you ever know if
  youre right?
• How accurate do we have to be?
• We can create data sets to test trees that we
  create and assume that it will then work in the
  real world

14
Data Sets

• JC Model
• Sites evolve independent
• Sites change with the same probability
• Changes are single character changes
• Ie. A -gt G or T -gt C
• The expectation of change is a Poisson variable
  ?(e)

15
More Data Sets

• K2P Model
• Based on JC Model
• Allows for probability of transitions to
  tranversions
• Its more likely for A and T to switch and G and
  C to switch
• Normally set to twice as likely

16
Data Use

• Using these data sets we can create our own
  evolution of data.
• Start with one ancestor and create evolutions
• Plug the evolutions back and see if you get what
  you started with

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Aspects of Trees

• Topology
• The method in which nodes are connected to each
  other
• Are we really connected to apes directly, or
  just linked long before we could be considered
  mammals?
• Distance
• The sum of the weighted edges to reach one node
  from another

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What can distance tell us?

• The distance between nodes IS the evolutionary
  distance between the nodes
• The distance between an ancestor and a
  leaf(present day object) can be interpreted as an
  estimate of the number of evolutionary steps
  that occurred.

19
Current Techniques

• Maximum Parsimony
• Minimize the total number of evolutionary events
• Find the tree that has a minimum amount of
  changes from ancestors
• Maximum Likelihood
• Probability based
• Which tree is most probable to occur based on
  current data

20
More Techniques

• Neighbor Joining
• Repeatedly joins pairs of leaves (or subtrees) by
  rules of numerical optimization
• It shrinks the distance matrix by considering two
  neighbors as one node

21
Learning Neighbor Joining

• It will become apparent later on, but lets learn
  how to do Neighbor Joining (NJ)

A
B
C
D
E
A
B
C
D
E
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NJ Part 1

• First start with a star tree

E
A
D
B
C
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NJ Part 2

• Combine the closest two nodes (from distance
  matrix)
• In our case it is node A and B at distance 3

E
A
D
B
C
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NJ Part 3

• Repeat this until you have added n-2 nodes (3)
• N-2 will make it a binary tree, so we only have
  to include one more node.

E
A
D
B
C
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Are we done?

• ML and MP, even in heuristic form take too long
  for large data sets
• NJ has poor topological accuracy, especially for
  large diameter trees
• We need something that works for large diameter
  trees and can be run fast.

26
Heres what we want

• Our Goal
• An Absolute Fast Converging Method
• ? is afc if, for all positive f,g, , on the
  Model M, there is a polynomial p such that, for
  all (T,?(e)) is in the set Mf,g on a set S of n
  sequences of length at least p(n) generated on T,
  we have Pr?(S) T gt 1- .
• Simply Lets make it in polynomial time within a
  degree of error.

27
A DCM - NJ Solution

• 2 Phase construction of a final phylogenetic tree
  given a distance matrix d.
• Phase 1 Create a set of plausible trees for the
  distance matrix
• Phase 2 Find the best fitting tree

28
Phase 1

• For each q in dij, compute a tree tq
• Let T tq q in dij

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Finding tq

• Step 1 Compute Thresh(d,q)
• Step 2 Triangulate Thresh(d,q)
• Step 3 Compute a NJ Tree for all maximal cliques
• Step 4 Merge the subtrees into a supertree

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What does that mean

• Breaking the problem up
• Create a threshold of diameters to break the
  problem into
• A bunch of smaller diameter trees (cliques)
• Apply NJ to those cliques
• Merge them back

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Finding tq (terms)

• Threshold Graph
• Thresh(d,q) is the threshold graph where (i,j) is
  an edge if and only if dij lt q.

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Threshold

• Lets bring back our distance matrix and create a
  threshold with q equal to d15 or the distance
  between A and E
• So q 67

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Distance Matrix

• Our old example matrix

A
B
C
D
E
A
B
C
D
E
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With q D15 67
C
47
A
67
D
63
B
E
16
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Triangulating

• A graph is triangulated if any cycle with four
  or more vertices has a chord
• That is, an edge joining two nonconsecutive
  vertices of the cycle.
• Our example is already triangulated, but lets
  look at another

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Triangulating
Lets say this is for q 5
10 and 15 would Not be in the graph
10
To triangulate this graph you add the edge
length 10.
15
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Maximal Cliques

• A clique that cannot be enlarged by the addition
  of another vertex.
• Recall our original threshold graph which is
  triangulated

38
Triangulated Threshold Graph

• Our old Graph

C
47
A
67
D
63
B
E
16
39
Clique

• Our maximal cliques would be
• A, B, E
• C, D

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Create Trees for the Cliques

• We have two maximal cliques, so we make two
  trees A, B, E and C, D
• How do we make these trees?
• Remember NJ?

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Tree A, B, E and C,D
A
E
B
C
D
42
Merge your separate trees together.

• Create one Supertree
• This is done by creating a minimum set of edges
  in the trees and calling that the backbone
• This is its own doctorial thesis, so lets do a
  little hand waving

43
That sounds like NP-hard!

• Computing Threshold is Polynomial
• Minimally triangulating is NP-hard, but can be
  obtained in polynomial time using a greedy
  heuristic without too much loss in performance.
• Maximal cliques is only polynomial if the data
  input is triangulated (which it is!).
• If all previous are done, creating a supertree
  can be done in polynomial time as well.

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Where are we now?

• We now have a finalized phylogeny created for
  from smaller trees in our matrix joined together
• Remember we started from all possible size of
  smaller trees.

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Phase 2

• Which one is right?
• Found using the SQS (Short Quartet Support)
  method
• Let T be a tree in S (made from part 1)
• Break the data into sets of four taxa
• A, B, C, D A, C, D, E A, B, D, E etc
• Reduce the larger tree to only hold one set
• These are called Quartets

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SQS - A Guide

• Q(T) is the set of trees induced by T on each set
  of four leaves.
• Let Qw (different Q) be a set of quartets with
  diameter less than or equal to w
• Find the maximum w where the quartets are
  inclusive of the nodes of the tree
• This w is the support of that tree

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SQS - Refrased

• Qw is the set of quartet trees which have a
  diameter lt w
• Support of T is the max w where Qw is a subset of
  Q(T)
• Support is our quality measure
• What are we exactly measuring?,

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Qw
A
B
D
D
E
C
A
B
A
B
C
D
A
B
C
D
E
E
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SQS Method

• Return the tree in which the support of that tree
  is the maximum.
• If more than one such tree exists return the tree
  found first.
• This is the tree with the smallest original
  diameter (remember from phase 1)

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How do we know were right?

• Compare it to the data set we created
• Look at Robinson-Foulds accuracy
• Remove one edge in the tree weve created.
• We now have two trees
• Is there anyway to create the same set of leaves
  by removing one edge in our data set?
• If no, add a point of error.
• Repeat this for all edges
• When the value is not zero then the trees are not
  identical

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Performance of DCM - NJ

• Outperforms NJ method at sequence lengths above
  4000 and with more taxa.

0.8
NJ
DCM-NJ
0.6
Error Rate
0.4
0.2
0
0
400
800
1600
1200
No. Taxa
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Improvements

• Improvement possibilities like in Phase 2
• Include test of Maximum Parsimony (MP)
• Try and minimize the overall size of the tree
• Test using statistical evidence
• Maximum Likelihood (ML)

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Performance gains

• Simply changing Phase 2 has massive gains in
  accuracy!
• DCM - NJ MP and DCM -NJ ML are VERY accurate
  for data sets greater than 4000 and are NOT NP
  hard.
• DCM - NJ MP finished its analysis on a 107
  taxon tree in under three minutes.

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Comparing Improvements
DCM-NJSQS
0.8
NJ
DCM-NJMP
HGT-FP
0.6
Error Rate
0.4
0.2
0
0
400
800
1600
1200
leaves