1. http://creativecommons.org/licenses/by-sa/2.0/

2. BNFO 602 Phylogenetics Usman Roshan

3. Phylogenetics Study of how species relate to each other “Nothing in biology makes sense, except in the light of evolution”, Theodosius Dobzhansky, Am. Biol. Teacher (1973) Rich in computational problems Fundamental tool in comparative bioinformatics

4. Why phylogenetics? Study of evolution –Origin and migration of humans –Origin and spead of disease Many applications in comparative bioinformatics –Sequence alignment –Motif detection (phylogenetic motifs, evolutionary trace, phylogenetic footprinting) –Correlated mutation (useful for structural contact prediction) –Protein interaction –Gene networks –Vaccine devlopment –And many more…

5. Phylogeny Problem TAGCCCATAGACTTTGCACAATGCGCTTAGGGCAT UVWXY U VW X Y

6. Bipartitions Phylogenies are equivalent to bipartitions

7. Topological differences

8. Phylogeny Problem Two main methodologies: –Alignment first and phylogeny second Construct alignment using one of the MANY alignment programs in the literature Do manual (eye) adjustments if necessary Apply a phylogeny reconstruction method Fast but biologically not realistic Phylogeny is highly dependent on accuracy of alignment (but so is the alignment on the phylogeny!) –Simultaneously alignment and phylogeny reconstruction Output both an alignment and phylogeny Computationally much harder Biologically more realistic as insertions, deletions, and mutations occur during the evolutionary process

9. First methodology Compute alignment (for now we assume we are given an alignment) Construct a phylogeny (two approaches) Distance-based methods –Input: Distance matrix containing pairwise statistical estimation of aligned sequences –Output: Phylogenetic tree –Fast but less accurate Character-based methods –Input: Sequence alignment –Output: Phylogenetic tree –Accurate but computationally very hard

10. Distance-based methods

11. Evolution on a single edge Poisson process –Number of changes in a fixed time interval t is independent of changes in any other non-overlapping time interval u –Number of changes in time interval t is proportional to the length of the interval –No changes in time interval of length 0 Let X be the number of nucleotide changes on a single edge. We assume X is a Poisson process Probability dictates that

12. Evolution on a single edge We want to compute (the probability of a nucleotide change on edge e) The probability of observing a change is just the sum of probabilities of observing k changes over all possible values of k (excluding even ones because those changes cannot be seen)

13. Expected number of nucleotide changes on a given edge is given by Key: is additive Evolution on a single edge

14. Additivity Assume we have a path of k edges and that p1, p2,…, pk are the probabilities of change on each edge of the path Using induction we can show that Multiplicative term is hard to deal with and does not easily decompose into a product or sum of pi’s

15. Additivity But the expected number of nucleotide changes on the path p is elegant

16. Evolutionary models Simple 0,1 alphabet evolutionary model –i.i.d. model –uniformly random root sequence Jukes-Cantor: –Uniformly random root sequence –i.i.d. model

17. General Markov Model –Uniformly random root sequence –i.i.d. model –For time reversible models Evolutionary models

18. Variation across sites Standard assumption of how sites can vary is that each site has a multiplicative scaling factor Typically these scaling factors are drawn from a Gamma distribution (or Gamma plus invariant)

19. Special issues Molecular clock: the expected number of changes for a site is proportional to time No-common-mechanism model: there is a random variable for every combination of edge and site

20. Evolutionary distance estimation

21. Estimating evolutionary distances For sequences A and B what is the evolutionary distance under the Jukes-Cantor model? –ACCTGTGGGTAACCACCC –ACCTGAGGGATAGGTCCG But we don’t know what is

22. Estimating evolutionary distances Assume nucleotide changes are Bernoulli trials (i.i.d. trials of success or failure) is probability of head in n Bernoulli trials (n is sequence length) Compute a maximum likelihood estimate for ACCTGTGGGTAACCACCC ACCTGAGGGATAGGTCCG 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1

23. Estimating evolutionary distance We want to find the value of p that maximizes the probability: Set dP/dp to 0 and solve for p to get

24. Estimating evolutionary distances = 5/18 Continuing in this manner we estimate for all pairs of sequences in the alignment We now have a distance matrix under a biologically sound evolutionary model ACCTGTGGGTAACCACCC ACCTGAGGGATAGGTCCG 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1

25. Distance methods

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

27. Additivity

28. UPGMA UPGMA is not additive but works for ultrametric trees. Takes O(n^3) time ADC B 3 3 3 10 A B C D A B CD 626 6 3

29. 1.Initialize n clusters where each cluster i contains the sequence i 2.Find closest pair of clusters i, j, using distances in matrix D 3.Make them neighbors in the tree by adding new node (ij), and set distance from (ij) to i and j as D ij /2 4.Update distance matrix D: for all clusters k do the following (ni and nj are size of clusters i and j respectively) 5.Delete columns and rows for i and j in D and add new ones corresponding to cluster (ij) with distances as computed above 6.Goto step 2 until only one cluster is left UPGMA

30. A B C D A B CD 626 6 ADC B 3 3 3 13 3

31. UPGMA Doesn’t work (in general) for non-ultrametric trees A D CB 10 3 33 3 A B C D A B CD 131626 1219 13

32. UPGMA UPGMA constructs incorrect tree here A B C D A B CD 131626 1219 13 BDA C 6 7.25 6

33. UPGMA Bipartition (BC,AD) is not in true tree BDA C 6 7.25 6 A D CB 10 3 33 3 True treeUPGMA tree

34. Neighbor joining 1.Additive and O(n^3) time 2.Initialization: same as UPGMA 3.For each species compute 4.Select i and j for which is minimum 5.Make them neighbors in the tree by adding new node (ij), and set distance from (ij) to i and j as

35. Neighbor joining 6.Update distance matrix D: for all clusters k do the following 7.Delete columns and rows for i and j in D and add new ones corresponding to cluster (ij) with distances as computed above 8.Go to 3 until two nodes/clusters are left

36. NJ NJ constructs the correct tree for additive matrices A D CB 10 3 33 3 A B C D A B CD 131626 1219 13

37. Simulation studies

38. 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

39. Statistical consistency As sequence lengths tend to infinity the distance estimation improves and eventually leads to the true additive matrix If a method like NJ is then applied we get the true tree. In practice, however, we have limited sequence length. Therefore we want to know how much sequence length a method requires to achieve low error

40. Convergence rates Can be studied experimentally or theoretically Theoretical results offer loose bounds Experiments (under simulation) provide more realistic bounds on sequence lengths

41. Sequence length requirements

43. Typical performance study

44. Sequence lengths for NJ Sequence lengths required to obtain 90% accuracy

45. Error rate of NJ

46. Improving sequence length requirements Later we will look at Disk-Covering Methods and study sequence length requirements of other methods (in addition to NJ)

47. Maximum Parsimony Character based method NP-hard (reduction to the Steiner tree problem) Widely-used in phylogenetics Slower than NJ but more accurate Faster than ML Assumes i.i.d.

48. Maximum Parsimony Input: Set S of n aligned sequences of length k Output: A phylogenetic tree T –leaf-labeled by sequences in S –additional sequences of length k labeling the internal nodes of T such that is minimized.

49. Maximum parsimony (example) Input: Four sequences –ACT –ACA –GTT –GTA Question: which of the three trees has the best MP scores?

50. Maximum Parsimony ACT GTTACA GTA ACA ACT GTA GTT ACT ACA GTT GTA

51. Maximum Parsimony ACT GTT GTA ACA GTA 1 2 2 MP score = 5 ACA ACT GTA GTT ACAACT 3 1 3 MP score = 7 ACT ACA GTT GTA ACAGTA 1 2 1 MP score = 4 Optimal MP tree

52. Maximum Parsimony: computational complexity ACT ACA GTT GTA ACAGTA 1 2 1 MP score = 4 Finding the optimal MP tree is NP-hard Optimal labeling can be computed in linear time O(nk)

53. Local search strategies Phylogenetic trees Cost Global optimum Local optimum

54. Local search for MP Determine a candidate solution s While s is not a local minimum –Find a neighbor s’ of s such that MP(s’)<MP(s) –If found set s=s’ –Else return s and exit Time complexity: unknown---could take forever or end quickly depending on starting tree and local move Need to specify how to construct starting tree and local move

55. Starting tree for MP Random phylogeny---O(n) time Greedy-MP

56. Greedy-MP takes O(n^2k^2) time

57. Local moves for MP: NNI For each edge we get two different topologies Neighborhood size is 2n-6

58. Local moves for MP: SPR Neighborhood size is quadratic in number of taxa Computing the minimum number of SPR moves between two rooted phylogenies is NP-hard

59. Local moves for MP: TBR Neighborhood size is cubic in number of taxa Computing the minimum number of TBR moves between two rooted phylogenies is NP-hard

60. Local optima is a problem

61. Iterated local search: escape local optima by perturbation Local optimum Local search

62. Iterated local search: escape local optima by perturbation Local optimum Output of perturbation Perturbation Local search

63. Iterated local search: escape local optima by perturbation Local optimum Output of perturbation Perturbation Local search

64. ILS for MP Ratchet Iterative-DCM3 TNT