Phylogenetics Eliran Avni, Reuven Cohen, Sagi Snir Presentation by - - PowerPoint PPT Presentation

Weighted Quartets Phylogenetics

Eliran Avni, Reuven Cohen, Sagi Snir

Presentation by Ashu Gupta

Motivation

• Computationally Difficult to analyze large datasets
• Solution?
• Divide and Conquer
• Step 1: Construct a set of subtrees (quartets) by

accurate phylogenetic methods

• Step 2:

Amalgamating the subtrees into a unified tree by a supertree method

Motivation (cont.)

Maximum Quartet Consistency

• Input:

A set of quartets Q Output: Tree T* such that the number of quartets in set Q which are satisfied by T* is maximized

• NP-Hard
• Need good heuristics to solve this problem

Quartet Max Cut

• Input:

A set of quartets Q, set of Taxa X

Output:

Tree T* (approximate solution to MQC)

• Divide and Conquer Amalgamation Technique
• Operates on the taxa set by partitioning it into parts, based on some optimization criterion
• Operate on the sub problems induced by each part
• Merges the sub-solutions into a complete solution.
• Each Partition represents a bipartition in the final tree
• Robust, doesn’t need all quartets

• At each recursion step taxon set X is partitioned into two parts P=(Y, X\Y)
• ab|cd Q is unaffected by a partition P, if all {a,b,c,d} are in one part of P.
• ab|cd is satisfied by P if some part contains precisely a and b, or some part contains precisely c and d.
• ab|cd is violated by P if some part contains a,c or a,d or b,c or b,d and the other part contains the
• ther two.
• Otherwise, some part contains only one of {a,b,c,d}

In this case ab|cd is deferred.

a b c d a b

c d

a d

b c

a b c

d

Quartet Max Cut (cont.)

• At every step of the algorithm, some quartets are satisfied, some

violated, and some continue to the next steps (i.e. either deferred or unaffected).

• Greedy Approach
• A plausible strategy is to maximize the ratio between satisfied and violated

quartets at every step.

• No Theoretical Guarantees!!

Quartet Max Cut (cont.)

• Given the set of quartets Q over a taxa set X, we build a graph GQ

=(X,E) with E as follows:

• For every ab|cd Q we add the 6 edges to E.
• The “crossing” edges ac, ad, bc, bd are good edges.
• The edges ab, cd are bad edges.

8

Q: GQ:

Bad Edges , Good Edges

Quartet Max Cut (cont.)

• A cut in GQ corresponds to a partition of the taxa set into two parts. Given

a cut C=(Y, X\Y) in the graph:

• A satisfied quartet contributes 4 good edges to the cut
• A deferred contributes 2 good edges and 1 bad edge
• A violated contributes 2 good edges and 2 bad edges
• We want to find a cut C* maximizing

C* = 𝒃𝒔𝒉𝒏𝒃𝒚𝑫 (|good edges| -  |bad edges|)

•  is dynamically chosen such that C* maximizes ρ(C*)=

|𝒉𝒑𝒑𝒆 𝒇𝒆𝒉𝒇𝒕| |𝒄𝒃𝒆 𝒇𝒆𝒉𝒇𝒕|

10

Q = { 12|34 , 13|45 }

GQ:

The cut {125}, {34} satisfies 12|34 but violates 13|45.

Quartet Max Cut (cont.)

• Problems?
• Each quartet has same weight
• What if we have confidence values for each quartet ?

(prior knowledge, confidence based on avg. branch length)

• Possible Solution?
• Consider only quartets having high confidence
• Loss of information
• BAD
• Need a better Amalgamation technique

Satisfies last 3 quartets Satisfies first 2 quartets

Which one is better?

Weighted Quartet Max Cut

• Intuition: Add confidence of quartets as weights to graph
• Build Graph GQ similar to QMC
• For each edge in GQ

Weight of edge = Weight of Mother Quartet

• We want to find a cut C* maximizing

C* = 𝒃𝒔𝒉𝒏𝒃𝒚𝑫 (|weight of good edges| -  |weight of bad edges|)

Definitions

• Weight of a Quartet given a model tree

𝒙𝒓=

(𝒆𝒊−𝒆𝒎) 𝒇(𝒆𝒊−𝒆𝒏)∗𝒆𝒊

𝑒𝑚, 𝑒𝑛, 𝑒ℎ represent the three pair wise sums

• Qfit
• Similarity measure between two trees based on quartets common to compared trees
• WQfit
• Novel similarity measure defined by the authors
• Takes into account both shared quartets and their weights to calculate similarity

Simulation

• Number of quartets used #𝑟𝑠𝑢 = 𝑜𝑙

where k = qrt-num-factor

• Rewire
• Choose a quartet randomly based on its confidence
• (low confidence) high probability of selection
• Randomly change the topology of the chosen quartet
• Weight of rewired quartets / Total weight = Ratio of rewire

Results

Results (cont.)

Results (cont.)

Results (cont.)

Results (cont.)

• Cynobacterial dataset (HGT is evident)
• Compared wQMC to embedded quartets method
• Embedded Quartets Method (Zhaxybayeva et al., 2006)
• Construct a tree for every gene
• Get induced quartet from every gene trees
• Get ML score for each quartet
• Remove low confidence quartets
• Run MRP to get super tree
• wQMC tree matched the Embedded Quartets

method

1128 genes, 214,729 quartets

Questions?

Thank You