03-511/711, 15-495/856 Course Notes - Oct 1, 09

Review of distance-based methods

• The matrix of observed distances, D[i,j], is obtained pairwise distances from a multiple sequence alignment. It must be corrected for multiple substitutions, as we discussed in the past two lectures.
• 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, D[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. The lenghth(s) of edge(s) connecting the subtrees are calculated and the distance matrix is updated. 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.
• How the distance matrix is updated.

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, D[], is ultrametric, then UPGMA will reconstruct the correct rooted tree in quadratic time.

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

• UPGMA will not give the correct tree even if D[] 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 D 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 D is additive, then NJ will reconstruct the correct unrooted tree in quadratic time.

Determining the root of the tree

If D is a ultrametric, then the root can be determined directly from the data as a consequence of the molecular clock hypothesis. The root is located at the midpoint of the longest pathway between two taxa. UPGMA does this automatically. For trees obtained using other algorithms, the root can be estimated using midpoint rooting after the tree is constructed, as long as the tree has branch lengths.

If D is not ultrametric, then additional information is needed. To root a tree one should add an outgroup to the data set. An outgroup is an taxon for which external information (eg. paleontological information, morphology, ...) is available that indicates that the outgroup branched off before all other taxa. For example, bear could be used as an outgroup in a canine phylogeny.

The outgroup should not be too closely related to the taxa in question. Nor should the outgroup be very distantly related to the taxa. If the rates of change do not differ greatly from one lineage to the next, then midpoint rooting may give a reasonable approximation.