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Building phylogenetic treesPowerPoint Presentation

Building phylogenetic trees

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Presentation Transcript

Overview

- Background
- Making a tree from pairwise distances;
- Parsimony;
- <break>;

- Assessing the trees: the bootstrap;
- Simultaneous alignment and phylogeny;
- Application: Phylip

Building phylogenetic trees

Background

- Phylogenetic tree: diagram showing evolutionary lineages of species/genes
- Trees are used:
- To understand lineage of various species
- To understand how various functions evolved
- To inform multiple alignments

Building phylogenetic trees

Phylogenetic tree approaches

- Distance:
- UPGMA
- Neighbour-joining

- Parsimony:
- Traditional parsimony
- Weighted parsimony

Building phylogenetic trees

Making a tree from pairwise distances

- Given a set of sequences you want to build a tree.
- Compute the distances dijbetween each pair i, j of the sequences.
- There are many different distance measures.
- Average distance between pairs of sequences from each cluster.

Building phylogenetic trees

UPGMA

- Unweighted Pair Group Method using arithmetic Averages.
- It works by clustering the sequences, at each stage combining two clusters and at the same time creating a new node in a tree, using a distance measure.

Building phylogenetic trees

Distance between points

- |Ci| and |Cj| denote the number of sequences in clusters i and j.

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Building phylogenetic trees

Distance between clusters

- Let Ckbe the union of clusters Ci and Cj,then dkl
- Where Cl is any other cluster.

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Building phylogenetic trees

Building the tree: UPGMA

Initialisation:

Assign each sequence i to its own cluster Ci,

Define one leaf of T for each sequence, and place at height zero.

Iteration:

Determine the two clusters i, j for which dij is minimal.

Define a new cluster k by , and define dkl for all l.

Define a node k with daughter nodes i an j, and place it at height dij /2.

Add k to the current clusters and remove i and j.

Terminiation:

When only two clusters i, j remain, place the root at height dij /2.

Building phylogenetic trees

UPGMA: Initialisation

Building phylogenetic trees

UPGMA: Iteration 1

Building phylogenetic trees

UPGMA: Iteration 2

Building phylogenetic trees

UPGMA: Iteration 3

Building phylogenetic trees

UPGMA: Terminiation

Building phylogenetic trees

Properties of UPGMA

- Molecular clock & ultrametric property of distances
- Additivity

Building phylogenetic trees

Properties of UPGMA:Molecular clock & ultrametric

- The molecular clock assumption: divergence of sequences is assumed to occur at the same rate at all points in the tree.
- If this does holds, then the data is said to be ultrametric.

Building phylogenetic trees

Properties of UPGMA:Additivity

- Given a tree, its edge lengths are said to be additive if the distance between any pair of leaves is the sum of the lengths of the edges on the path connecting them.

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Building phylogenetic trees

Neighbour-joining

- N-j constructs a tree by iteratively joining subtrees (like UPGMA).
- Produces an unrooted tree.
- Doesn’t make the molecular clock assumption, therefore the ultrametric property does not hold.

Building phylogenetic trees

Distances in Neighbour-joining

- Given a new internal nodek, the distance to another node m is given by:

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Building phylogenetic trees

Distances in Neighbour-joining

- Generalizing this so that the distance to all other leaves are taken into account:
- Where
- And |L| denotes the size of the set L of leaves.

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Building phylogenetic trees

Building the tree:Neighbour-joining

Initialisation:

Define T to be the set of leaf nodes, one for each given sequence, and put L=T.

Iteration:

Pick a pair i, j in L for which defined by is minimal.

Define a new node k and set , for all m in L.

Add k to T with edges of lengths ,

joining k to i and j, respectively.

Remove i and j from L and add k.

Termination:

When L consists of two leaves i and j add the remaining edge between i and j, with length dij.

Building phylogenetic trees

Candidateroot

Rooting treesm

- Finding a root in an unrooted tree is sometimes accomplished by using an outgroup:
- A species known to be more distantly related to remaining species than they are to each other

- The point where the outgroup joins the rest of the tree is the best candidate for root position

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Building phylogenetic trees

Comments on distance based methods

- If the given data is ultrametric (and these distances represent real distances), then UPGMA will identify the correct tree.
- If the data is additive (and these distances represent real distances), then Neighbour-joining will identify the correct tree.
- Otherwise, the methods may not recover the correct tree, but they may still be reasonable heuristics.

Building phylogenetic trees

Phylogenetic tree approaches

- Distance:
- UPGMA
- Neighbour-joining

- Parsimony:
- Traditional parsimony
- Weighted parsimony

Building phylogenetic trees

Parsimony

- Most widely used tree building algorithm(?).
- Finds the tree that explains the data with a minimal number of changes.
- Instead of building atree, it assigns a cost to a given tree.
- Two components of the parsimony algorithm can be distinguished:
- The computation of a cost for a given tree;
- A search through all trees, to find the overall minimum of this cost.

Building phylogenetic trees

Parsimony example

- Given the following sequences: AAG,AAA,GGA,AGA.
- Several trees could explain the phylogeny

Building phylogenetic trees

Traditional Parsimony

- Count the number of substitutions
- At each node keep:
- a list of minimal cost residues
- the current cost

- Post-order traversal of the tree

Building phylogenetic trees

Traditional Parsimony

Initialisation:

Set current cost C=0 and k =2n-1, the number of the root node.

Recursion: To obtain the set Rk:

If k is a leaf node:

Set

If k is not a leaf node:

Compute Ri , Rj for the daughter i, j of k, and set if this intersection is not empty, or else set and increment C.

Termination:

Minimal cost of tree = C.

Building phylogenetic trees

Weighted Parsimony

- Extension of the traditional parsimony.
- Adds a cost function S(a,b) for each substitution of a by b.
- Post-order traversal of the tree
- Aim is now to minimize the cost.

Building phylogenetic trees

Weighted Parsimony

Initialisation:

Set k =2n-1, the number of the root node

Recursion: Compute Sk(a) for all a as follows:

If k is a leaf node:

Set , otherwise

If k is not a leaf node:

Compute Si(a), Sj(a) for all a at the daughter i, j and define

Termination:

Minimal cost of tree = minaS2n-1(a).

Building phylogenetic trees

Break

- Questions so far?
- After the break:
- Assessing the trees: the bootstrap;
- Simultaneous alignment and phylogeny;
- Application: Phylip

Building phylogenetic trees

Branch and bound

- Parsimony itself can not build a tree!
- Using simple enumeration methods the number of trees become very large very fast.
- How to build the trees?
- Stochastically
- Branch and bound

Building phylogenetic trees

Branch and bound

- B&B uses the parsimony algorithm.
- It guarantees to find the overall best tree.
- It systematically builds trees by increasing the number of leaves.
- Abandons a particular avenue of tree building whenever the current incomplete tree (T*) has a cost(T*)>cost(Tmin).

Building phylogenetic trees

The Bootstrap

- A measure how much a tree should be trusted.
- Use the bootstrap as a method of assessing the significance of some phylogenetic feature.

Building phylogenetic trees

The Bootstrap (2)

- The bootstrap works as follows:
- Given a dataset of an alignment of sequences.
- Generate an artificial dataset of the same size as the original dataset by picking columns from the alignment at random with replacement.
- Apply the tree building algorithm to this artificial dataset.
- Repeat selection and tree building procedure n times.
- The feature with which a chosen phylogenetic features appears is taken to be a measure of the confidence we can have in this feature.

Building phylogenetic trees

Simultaneous alignment and phylogeny

- Simultaneously aligning sequences and finding a plausible phylogeny:
- Sankoff & Cedergren’s gap-substitution algorithm;
- Hein’s affine cost algorithm.

- Both find an optimal alignment given a tree.

Building phylogenetic trees

Sankoff & Cedergren’s gap-substitution algorithm

- Guarantees to find ancestral sequences, and alignments of them and the leaf sequences.
- It uses a character-substitution model of gaps
- Together this minimizes a tree-based parsimony-type cost.
- The algorithm is a combination of two known methods:
- Dynamic programming method (Chapter 6);
- Weighted Parsimony algorithm.

Building phylogenetic trees

Hein’s affine cost algorithm

- It uses affine gap penalties.
- Faster than the Sankoff & Cedergren algorithm.
- The aim is to find sequences z at a given node aligned to both of the sequences x and y at the daughter nodes satisfying:
- Where S is the total cost for a given alignment of two sequences. (mismatch cost =1 and 0 otherwise)

Building phylogenetic trees

Hein’s affine cost algorithm

- Compared to equation (2.16) (alignment with affine gap scores) here the algorithm searches for the minimal cost path.
- The affine gap cost for a gap of length k isd+(k-1)e, where e<=d.

Building phylogenetic trees

Hein’s affine cost algorithm

- Find the zfor which is minimal.
- From the matrix follows:
- C - - A C -
- C A C - - -

- CAC could be possible z.

CAC(?)

CAC

CTCACA

Building phylogenetic trees

Which zcould serve best as ancestor?

Hein’s affine cost algorithmCAC(?)

CACACA(?)

CAC

CTCACA

CAC

CTCACA

CACAC(?)

CAC

CTCACA

Building phylogenetic trees

Sequence graph

- Follow a path through the dynamic programming matrix.
- Derive a graph from this matrix.
- Whenever a cell is used by an optimal path a vertex is added to the graph.

Building phylogenetic trees

Dynamic Programming matrix:TAC – Graph 3

Building phylogenetic trees

Ancestors

- Possible ancestral sequences for the leaf sequences TAC, CAC and CTCACA given the tree shown.
- Derived from the sequence graphs.

CAC

1

CAC

TAC

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CAC

CTCACA

Building phylogenetic trees

Limitations of Hein’s model

- Hein’s algorithm takes the minimal cost sequences at each node upward.
- This can fail to give the overall optimum.
- Suppose the cost for a gap of length k is:
- 13+3(k-1)

- Mismatch:
- 4

- Suppose the leaves G and GTT.

Building phylogenetic trees

Limitations of Hein’s model

- A eligible ancestor of G and GTT would be themselves, since they both have a cost of 13+3=16.
- GT would not be eligible because of the total cost of 2*13=26.
- Now we want to branch to the ancestor of G and GTT and there is a third leave GT.
- The total cost for ineligible GT would be lower than for either G or GTT.

Building phylogenetic trees

Application: PHYLIP (Phylogeny Inference Package)

- Many features, among:
- Traditional (unrooted) parsimony
- Branch and bound to find all most parsimonious trees

Building phylogenetic trees

Application: PHYLIP

- Test dataset:
Jurgen AACGUGGCCAAAU

Alpha ACCGCCGCCAAAU

Beta AAGGUCGCCAAAC

Gamma CAUUUCGUCACAA

Delta GGUAUCUCGGCCU

Epsilon GAAAUCUCGAUCC

Richard GGGCUCUCGGCUC

Building phylogenetic trees

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