03-511/711, 15-495/856 Course Notes

03-511/711, 15-495/856 Course Notes - Oct 05, 2006

Distance-based methods

The matrix of observed distances, O[i,j], is obtained pairwise distances from a multiple sequence alignment. It must be corrected for multiple substitutions.
Fitting distances to a tree
- Additive matrices
  - distances fit a tree
  - satisfy the 4-point condition
- Ultrametric matrices
  - distance fit a rooted tree where all paths to the root are of equal length
  - satisfy the 3-point condition
  - rate of mutation is the same in all lineages
  - branch lengths equal time

Greedy methods for distance-based phylogeny reconstruction

Taxa are points in a metric space with pairwise distances, O[i,j]. Tree building is equivalent to hierachical clustering of these points.

Both of these greedy algorithms maintain a forest of subtrees, beginning with the set of singleton trees (i.e., trees with one leaf and no edges). At each iteration, the algorithm merges two neighboring subtrees in the forest. This step is repeated until only one tree remains - the final result.

The algorithms differ in

How neighbors to be merged are identified.
How the branch lengths are computed.

Unweighted Paired Group Method with Average Means (UPGMA)

The UPGMA algorithm is a variant of average linkage. UPGMA is based on the molecular clock assumption. The consequences of this assumption are that

At each step, the two closest taxa are selected as neighbors.
The height of the least common ancestor of any pair of leaves is half the distance between the leaves.
If a distance matrix, O[], is an ultrametric, then UPGMA will reconstruct the correct rooted tree in quadratic time.

However, if the assumption is violated (i.e., if O is a not ultrametric), then

UPGMA will not give the correct tree even if O[] is additive.
The tree that results can have the incorrect topology and/or the incorrect branch lengths.
Errors in topology can occur because UPGMA selects pairs of taxa with minimum distance as neighbors, leading to the wrong tree when the closest taxa are not adjacent in the true tree.
Errors in the branch lengths can occur because UPGMA assumes that the internal node joining a pair of leaves is always equidistant between them.

Neighbor Joining (NJ)

The NJ algorithm deals with this problem by correcting for variations in the rate of change. The "corrected" distance between a pair of nodes is calculated by subtracting the average of the distances to all other leaves.

Thm:: If O is additive, the pair of taxa that minimimize this "corrected" distance matrix are neighbors in the true tree.
Proof:: Durbin et al., 7.8

If O is additive, then NJ will reconstruct the correct unrooted tree in quadratic time.

Distance-based phylogeny reconstruction when the observed distance do not fit a tree

UPGMA and Neighbor Joining (NJ) are greedy algorithms that will reconstruct the correct tree in polynomial time when O[i,j], the matrix of observed inter-taxon distances, is ultrametric or additive, respectively.

How should you reconstruct a tree when O[i,j] is not additive? NJ can be used as a greedy heuristic. It is not guaranteed to give the correct tree, but if O[i,j] does not deviate too far from additive it may give a tree which is a good hypothesis for the evolutionary events that occurred.

Alternatively, one may use distance-based exhaustive or heuristic search. The basic approach is as follows

For each tree topology, t, with
leaves
- Estimate the branch lengths.
- Score the tree.
Select the tree with optimal score.

This problem is NP-complete. Since the number of trees grows very rapidly with the number of taxa, exhaustive search can only be used for small data sets.

The following questions remain:

How do we estimate the branch lengths?
How do we score the tree?

Branch Lengths

We select branch lengths that minimize the error in fitting the tree metric, T[i,j], to O[i,j]. There are a number of ways to measure this error. Many take the following approach:

Q =	_{k k}
	∑ ∑ (O[i,j] - T[i,j])²	(1)
	_i=1 j=i+1

where

Q is error fitting distances to tree
k is the number of taxa
O[i,j] is the observed pairwise distance between taxon i and taxon j
T[i,j] is the distance between i and j on the tree

To actually compute the branch lengths requires considerable computational effort. We need to find edge lengths x = x₁, x₂, ... such that the error Q given in Equation (1) is minimized. We do this by solving the equation Q' = 0. This means that we must solve a system of equations for each tree topology. Sometimes the following constraints are imposed:

The edge lengths are greater than zero: x_i > 0 .
The tree distances are greater than the observed distances: T[i,j]≥ O[i,j]. This is because multiple substitutions tend to cause us to underestimate intertaxon distances.

Scoring the tree

There are several approaches to scoring the tree. One approach is to score the tree according to its error, as given in Equation 1. In this case, the tree with the minimum error is sought.

Another frequently used approach is Minimum Evolution. In this method, the branch lengths are fitted using ordinary least squares (Equation 1). The score of the tree is the length of the tree; that is, by summing the lengths of all branches in the tree.

Rzhetsky and Nei (1993) claim that minimum evolution performs well on simulated trees and that under certain conditions the true tree minimizes the tree length score. However, Gascuel, 2000 reported that minimum evolution did not work much better than NJ in his simulation studies.

Evaluating how well the data supports a given tree

Bootstrapping, Branches and Partitions

Last modified: October 5th, 2006.
Maintained by Dannie Durand and Annette McLeod (durand@cs.cmu.edu).