Solving phylogenetic trees
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Solving Phylogenetic Trees. Benjamin Loyle March 16, 2004 Cse 397 : Intro to MBIO. Table of Contents. Problem & Term Definitions A DCM*-NJ Solution Performance Measurements Possible Improvements. Phylogeny. From the Tree of the Life Website, University of Arizona. Orangutan. Human.

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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


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


-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

DNA Sequence Evolution

Benjamin 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


Distance Matrix

  • Lets start with an example matrix

A

B

C

D

E

A

B

C

D

E

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 1

    • First start with a “star tree”

    E

    A

    D

    B

    C

    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


    Phase 1

    • For each q in {dij}, compute a tree tq

    • Let T = { tq : q in {dij} }

    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


    Distance Matrix

    • Our old example matrix

    A

    B

    C

    D

    E

    A

    B

    C

    D

    E

    Benjamin Loyle 2004 Cse 397


    With q = D15 = 67

    C

    47

    A

    67

    D

    63

    B

    E

    16

    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


    5

    W

    X

    5

    5

    Y

    Z

    5

    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

    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


    Triangulated Threshold Graph

    • Our old Graph

    C

    47

    A

    67

    D

    63

    B

    E

    16

    Benjamin Loyle 2004 Cse 397


    Clique

    Our maximal cliques would be:

    {A, B, E}

    {C, D}

    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


    Tree {A, B, E} and {C,D}

    A

    E

    B

    C

    D

    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


    Qw =

    A

    B

    D

    D

    E

    C

    A

    B

    A

    B

    C

    D

    E

    A

    B

    C

    D

    E

    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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