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### Solving Phylogenetic Trees

Benjamin Loyle

March 16, 2004

Cse 397 : Intro to MBIO

Benjamin Loyle 2004 Cse 397

Table of Contents

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

Benjamin Loyle 2004 Cse 397

Phylogeny

From the Tree of the Life Website,University of Arizona

Orangutan

Human

Gorilla

Chimpanzee

Benjamin Loyle 2004 Cse 397

AAGACTT

-2 mil yrs

AAGGCCT

AAGGCCT

TGGACTT

TGGACTT

-1 mil yrs

AGGGCAT

TAGCCCT

AGCACTT

AGGGCAT

TAGCCCT

AGCACTT

today

AGGGCAT

TAGCCCA

TAGACTT

AGCACAA

AGCGCTT

DNA Sequence EvolutionBenjamin Loyle 2004 Cse 397

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.

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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!

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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.

Benjamin Loyle 2004 Cse 397

Lets make it simple (constrain the input)

- Lets keep the distance between nodes within a certain limit
- From F -> 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.

Benjamin Loyle 2004 Cse 397

ERROR?!?!!?

- All trees are inferred, how do you ever know if you’re 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

Benjamin Loyle 2004 Cse 397

Data Sets

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

Benjamin Loyle 2004 Cse 397

More Data Sets

- K2P Model
- Based on JC Model
- Allows for probability of transitions to tranversions
- It’s more likely for A and T to switch and G and C to switch
- Normally set to twice as likely

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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.

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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.

Benjamin Loyle 2004 Cse 397

Here’s 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] > 1- €.
- Simply: Lets make it in polynomial time within a degree of error.

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

Finding tq (terms)

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

W

X

5

5

Y

Z

5

TriangulatingLets 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

Benjamin Loyle 2004 Cse 397

Maximal Cliques

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

Benjamin Loyle 2004 Cse 397

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?

Benjamin Loyle 2004 Cse 397

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 it’s own doctorial thesis, so lets do a little hand waving

Benjamin Loyle 2004 Cse 397

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.

Benjamin Loyle 2004 Cse 397

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.

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

SQS - Refrased

- Qw is the set of quartet trees which have a diameter <= 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?,

Benjamin Loyle 2004 Cse 397

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)

Benjamin Loyle 2004 Cse 397

How do we know we’re right?

- Compare it to the data set we created
- Look at Robinson-Foulds accuracy
- Remove one edge in the tree we’ve 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

Benjamin Loyle 2004 Cse 397

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

1200

1600

No. Taxa

Benjamin Loyle 2004 Cse 397

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)

Benjamin Loyle 2004 Cse 397

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.

Benjamin Loyle 2004 Cse 397

Comparing Improvements

DCM-NJ+SQS

0.8

NJ

DCM-NJ+MP

HGT-FP

0.6

Error Rate

0.4

0.2

0

0

800

400

1200

1600

# leaves

Benjamin Loyle 2004 Cse 397

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