CSCE555 Bioinformatics

1. CSCE555 Bioinformatics Lecture 13 Phylogenetics II Meeting: MW 4:00PM-5:15PM SWGN2A21 Instructor: Dr. Jianjun HuCourse page: http://www.scigen.org/csce555 HAPPY CHINESE NEW YEAR University of South Carolina Department of Computer Science and Engineering 2008 www.cse.sc.edu.

2. Outline • Review For Exams • Data for Phylogenetic Tree inference • Classification of Tree inference approaches • Neighbor-joining algorithm • Parsimony-based tree reconstruction • Least Square Best-fit reconstruction

3. Midterm, Midterm • How to review: read slides and textbooks, especially CG book. • Format of problems: examples • Brief questions: what is the difference between global alignment and local alignment? • calculation: build a HMM model for a multiple seq alignment • Definition: blasting, Motif, ORF

4. Covered Topics • Understand: concepts, algorithm ideas, tools • Sequencing/blasting • Gene finding • Alignment algorithms and applications • DNA motif search • HMM profiles • Gene prediction algorithms • Promoter predictions • Comparative genomics • ……

5. Phylogenetic Reconstruction • There are essentially two types of data for phylogenetic tree estimation: • Distance data, usually stored in a distance matrix, e.g. DNA×DNA hybridisation data, morphometric differences, immunological data, pairwise genetic distances • Character data, usually stored in a character array; • e.g. multiple sequence alignment of DNA sequences, morphological characters. Characters Distances 1 1 2 3 4 5 6 7 8 9 0 A A 1 0 0 0 1 1 0 1 1 1 B B 0 1 0 0 1 1 1 1 1 1 Taxa C Taxa C 0 0 1 0 0 0 1 1 1 1 D D 0 0 0 1 0 0 0 0 0 1 E E 0 0 0 0 0 0 0 0 0 0

6. Phylogenetic Reconstruction • Given the huge number of possible trees even for small data sets, we have two options: • Build one according to some clustering algorithm • Assign a “goodness of fit” criterion (an objective function) and find the tree(s) which optimise(s) this criterion

7. Phylogenetic Reconstruction Type of Data Nucleotide Distances Sites UPGMA Clustering Algorithm Neighbor-Joining Tree Building Method Maximum Parsimony Optimality Minimum Criterion Evolution Maximum Likelihood CS369 2007

8. Phylogenetic Methods Many different procedures exist. Three of the most popular: Neighbor-joining • Minimizes distance between nearest neighbors Maximum parsimony • Minimizes total evolutionary change Maximum likelihood • Maximizes likelihood of observed data

9. Distance based tree Construction Given a set of species (leaves in a supposed tree), and distances between them – construct a phylogeny which best “fits” the distances. Orc: ACAGTGACGCCCCAAACGT Elf: ACAGTGACGCTACAAACGT Dwarf: CCTGTGACGTAACAAACGA Hobbit: CCTGTGACGTAGCAAACGA Human:CCTGTGACGTAGCAAACGA Orc Elf Dwarf Hobbit Human

10. Distance Matrix • Given n species, we can compute the n x n distance matrixDij • Dij may be defined as the edit distance between a gene in species i and species j, where the gene of interest is sequenced for all n species. • Dij can also be any other feature-based distances

11. Distances in Trees • Edges may have weights reflecting: • Number of mutations on evolutionary path from one species to another • Time estimate for evolution of one species into another • In a tree T, we often compute dij(T) - the length of a path between leaves i and j

12. Distances in Trees • Edges may have weights reflecting: • Number of mutations on evolutionary path from one species to another • Time estimate for evolution of one species into another • In a tree T, we often compute dij(T) - the length of a path between leaves i and j

13. j i Distance in Trees: an Exampe d1,4 = 12 + 13 + 14 + 17 + 12 = 68

14. Fitting Distance Matrix • Given n species, we can compute the n x n distance matrixDij • Evolution of these genes is described by a tree that we don’t know. • We need an algorithm to construct a tree that best fits the distance matrix Dij

15. Tree reconstruction for any 3x3 matrix is straightforward We have 3 leaves i, j, k and a center vertex c Reconstructing a 3 Leaved Tree Observe: dic + djc = Dij dic + dkc = Dik djc + dkc = Djk

16. dic + djc = Dij + dic + dkc = Dik 2dic + djc + dkc = Dij + Dik 2dic + Djk = Dij + Dik dic = (Dij + Dik – Djk)/2 Similarly, djc = (Dij + Djk – Dik)/2 dkc = (Dki + Dkj – Dij)/2 Reconstructing a 3 Leaved Tree

17. Trees with > 3 Leaves • An tree with n leaves has 2n-3 edges • This means fitting a given tree to a distance matrix D requires solving a system of “n choose 2” equations with 2n-3 variables • This is not always possible to solve for n > 3

18. Additive Distance Matrices Matrix D is ADDITIVE if there exists a tree T with dij(T) = Dij NON-ADDITIVE otherwise

19. Distance Based Phylogeny Problem • Goal: Reconstruct an evolutionary tree from a distance matrix • Input: n x n distance matrix Dij • Output: weighted tree T with n leaves fitting D • If D is additive, this problem has a solution and there is a simple algorithm to solve it

20. Find neighboring leavesi and j with parent k Remove the rows and columns of i and j Add a new row and column corresponding to k, where the distance from k to any other leaf m can be computed as: Using Neighboring Leaves to Construct the Tree Compress i and j into k, iterate algorithm for rest of tree Dkm = (Dim + Djm – Dij)/2

21. Finding Neighboring Leaves • To find neighboring leaves we simply select a pair of closest leaves.

22. Finding Neighboring Leaves • To find neighboring leaves we simply select a pair of closest leaves. • WRONG

23. Finding Neighboring Leaves • Closest leaves aren’t necessarily neighbors • i and j are neighbors, but (dij= 13) > (djk = 12) • Finding a pair of neighboring leaves is • a nontrivial problem!

24. Neighbor Finding: Seitou & Nei algorithm (1987) Definitions Theorem (Saitou & Nei)Assume all edge weights are positive. If D(i,j) is minimal (among all pairs of leaves), then i and j are neighboring leaves in the tree.

25. Neighbor Joining Algorithm • In 1987 Naruya Saitou and Masatoshi Nei developed a neighbor joining algorithm for phylogenetic tree reconstruction • Finds a pair of leaves that are close to each other but far from other leaves: implicitly finds a pair of neighboring leaves • Advantages: works well for additive and other non-additive matrices, it does not have the flawed molecular clock assumption

26. Neighbor-joining • Guaranteed to produce the correct tree if distance is additive • May produce a good tree even when distance is not additive Step 1: Finding neighboring leaves Define Dij = dij – (ri + rj) Where 1 ri = –––––k dik |L| - 2 1 3 0.1 0.1 0.1 0.4 0.4 4 2

27. Algorithm: Neighbor-joining Initialization: Define T to be the set of leaf nodes, one per sequence Let L = T Iteration: • Pick i, j s.t. Dij is minimal • Define a new node k, and set dkm = ½ (dim + djm – dij) for all m  L • Add k to T, with edges of lengths dik = ½ (dij + ri – rj) • Remove i, j from L; • Add k to L Termination: When L consists of two nodes, i, j, and the edge between them of length dij

28. Rooting a tree, and definition of outgroup Neighbor-joining produces an unrooted tree How do we root a tree between N species using n-j? An outgroup is a species that we know to be more distantly related to all remaining species, than they are to one another Example: Human, mouse, rat, pig, dog, chicken, whale Which one is an outgroup? Outgroup can act as a root 1 4 3 2

29. Neighbor Joining Algorithm-Widely Used • Applicable to matrices which are not additive • Known to work good in practice • The algorithm and its variants are the most widely used distance-based algorithms today.

30. Maximum Parsimony Method for Tree Inference A Character-based method Input: h sequences (one per species), all of length k. Goal: Find a tree with the input sequences at its leaves, and an assignment of sequences to internal nodes, such that the total number of substitutions is minimized. • Two sub-problems: • Find the parsimony cost of a given tree (easy) • Search through all tree topologies (hard)

31. AAA AAA AAA 2 1 1 GGA AGA AAG AAA Total #substitutions = 4 Example Input: four nucleotide sequences: AAG, AAA, GGA, AGA taken from four species. By the parsimony principle, we seek a tree that has a minimum total number of substitutions of symbols between species and their originator in the phylogenetic tree. Here is one possible tree.

32. Least Squares Distance Phylogeny Problem • If the distance matrix D is NOT additive, then we look for a tree T that approximates D the best: Squared Error : ∑i,j (dij(T) – Dij)2 • Squared Error is a measure of the quality of the fit between distance matrix and the tree: we want to minimize it. • Least Squares Distance Phylogeny Problem: finding the best approximation tree T for a non-additive matrix D (NP-hard).

33. Search through tree topologies: Branch and Bound Observation: adding an edge to an existing tree can only increase the parsimony cost Enumerate all unrooted trees with at most n leaves: [i3][i5][i7]……[i2N–5]] where each ik can take values from 0 (no edge) to k At each point keep C = smallest cost so far for a complete tree Start B&B with tree [1][0][0]……[0] Whenever cost of current tree T is > C, then: • T is not optimal • Any tree with more edges containing T, is not optimal: Increment by 1 the rightmost nonzero counter

34. Comparison of Methods

35. Summary • Category of phylogenetic inference algorithms • Neighbor-joining algorithm

36. Acknowledgement • Anonymous authors