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Solving Phylogenetic Trees

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Solving Phylogenetic Trees

Benjamin Loyle

March 16, 2004

Cse 397 : Intro to MBIO

Benjamin Loyle 2004 Cse 397

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

Benjamin Loyle 2004 Cse 397

From the Tree of the Life Website,University of Arizona

Orangutan

Human

Gorilla

Chimpanzee

Benjamin Loyle 2004 Cse 397

-3 mil yrs

AAGACTT

-2 mil yrs

AAGGCCT

AAGGCCT

TGGACTT

TGGACTT

-1 mil yrs

AGGGCAT

TAGCCCT

AGCACTT

AGGGCAT

TAGCCCT

AGCACTT

today

AGGGCAT

TAGCCCA

TAGACTT

AGCACAA

AGCGCTT

Benjamin Loyle 2004 Cse 397

- 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

- 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

- 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

- 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

- 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

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 start with an example matrix

A

B

C

D

E

A

B

C

D

E

Benjamin Loyle 2004 Cse 397

- 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

- 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

- 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

- 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

- 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

- 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?”

- The sum of the weighted edges to reach one node from another

Benjamin Loyle 2004 Cse 397

- 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

- 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

- 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

- 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

- First start with a “star tree”

E

A

D

B

C

Benjamin Loyle 2004 Cse 397

- 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

- 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

- 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

- 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.

- An “Absolute Fast Converging” Method

Benjamin Loyle 2004 Cse 397

- 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

- For each q in {dij}, compute a tree tq
- Let T = { tq : q in {dij} }

Benjamin Loyle 2004 Cse 397

- 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

- Breaking the problem up
- Create a threshold of diameters to break the problem into
- A bunch of smaller diameter trees (cliques)

- Create a threshold of diameters to break the problem into
- Apply NJ to those cliques
- Merge them back

Benjamin Loyle 2004 Cse 397

- 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

- 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

- Our old example matrix

A

B

C

D

E

A

B

C

D

E

Benjamin Loyle 2004 Cse 397

C

47

A

67

D

63

B

E

16

Benjamin Loyle 2004 Cse 397

- 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

5

W

X

5

5

Y

Z

5

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

Benjamin Loyle 2004 Cse 397

- 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

- Our old Graph

C

47

A

67

D

63

B

E

16

Benjamin Loyle 2004 Cse 397

Our maximal cliques would be:

{A, B, E}

{C, D}

Benjamin Loyle 2004 Cse 397

- 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

A

E

B

C

D

Benjamin Loyle 2004 Cse 397

- 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

- 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

- 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

- 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

- 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

- 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

Qw =

A

B

D

D

E

C

A

B

A

B

C

D

E

A

B

C

D

E

Benjamin Loyle 2004 Cse 397

- 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

- 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

- Remove one edge in the tree we’ve created.

Benjamin Loyle 2004 Cse 397

- 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

- 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

- 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

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