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Distance methods

Distance methods. Distance methods. UPGMA: similar to hierarchical clustering but not additive Neighbor-joining: more sophisticated and additive What is additivity?. Additivity. UPGMA. UPGMA is not additive but works for ultrametric trees. Takes O(n^3) time. B. A. C. D. A. 6. 26.

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Distance methods

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  1. Distance methods

  2. Distance methods • UPGMA: similar to hierarchical clustering but not additive • Neighbor-joining: more sophisticated and additive • What is additivity?

  3. Additivity

  4. UPGMA UPGMA is not additive but works for ultrametric trees. Takes O(n^3) time B A C D A 6 26 26 10 10 26 26 B 6 C 3 3 3 3 D A C D B

  5. UPGMA • Initialize n clusters where each cluster i contains the sequence i • Find closest pair of clusters i, j, using distances in matrix D • Make them neighbors in the tree by adding new node (ij), and set distance from (ij) to i and j as Dij/2 • Update distance matrix D: for all clusters k do the following (ni and nj are size of clusters i and j respectively) • Delete columns and rows for i and j in D and add new ones corresponding to cluster (ij) with distances as computed above • Goto step 2 until only one cluster is left

  6. UPGMA B A C D 13 13 A 6 32 32 32 32 B 3 6 3 C 3 3 D A C D B

  7. UPGMA Doesn’t work (in general) for non ultrametric trees B A C D 3 3 A 13 19 26 3 3 12 19 B 10 10 B C 13 C D D A

  8. UPGMA UPGMA constructs incorrect tree here 7.25 B A C D 7.25 A 13 26 19 7.25 7.25 12 19 B 6 6 13 C B A D C D

  9. UPGMA Bipartition (BC,AD) is not in true tree 7.25 3 3 3 3 7.25 7.25 7.25 10 10 B C 6 6 D A B A D C True tree UPGMA tree

  10. Neighbor joining • Additive and O(n^3) time • Initialization: same as UPGMA • For each species compute • Select i and j for which is minimum • Make them neighbors in the tree by adding new node (ij), and set distance from (ij) to i and j as

  11. Neighbor joining • Update distance matrix D: for all clusters k do the following • Delete columns and rows for i and j in D and add new ones corresponding to cluster (ij) with distances as computed above • Go to 3 until two nodes/clusters are left

  12. NJ NJ constructs the correct tree for additive matrices B A C D 3 3 A 13 19 26 3 3 12 19 B 10 10 B C 13 C D D A

  13. Simulation studies

  14. Simulation studies • The true evolutionary tree is never known in practice. Simulation allows us to study accuracy of methods under biologically realistic scenarios • Mathematics behind the phylogenetics is often complex and challenging. Simulation allows us to study algorithms when not possible theoretically and also examine algorithm performance under various conditions such as different evolutionary rates, sequence lengths, or numbers of taxa

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