Bioinformatics - Lecture 7

Transcription

1 Bioinformatics - Lecture 7 Louis Wehenkel Department of Electrical Engineering and Computer Science University of Liège Montefiore - Liège - November 20, 2007 Find slides: lwh/ibioinfo/ Louis Wehenkel GBIO Bioinformatique (1/16)

2 Chapter 7. Phylogenetic analysis Louis Wehenkel GBIO Bioinformatique (2/16)

3 Phylogenetic analysis Objective: From a set of DNA sequences which have been obtained from a certain individuals in a population, derive the parent-child relationship among these individuals. Derive time since divergence from common ancestor. Derive a tree, highlighting successive points of divergence in time. Applications: Understand virus evolution (short time intervals). Understand human population migrations (intermediate time intervals). Understand evolution among species (long time intervals). Louis Wehenkel GBIO Bioinformatique (3/16)

4 Mathematical notion of graph General notion of (finite, unweighted) graph: Finite set of nodes N and of branches B N N. Self-loop: arc of type (i, i). In this chapter we consider graphs without self-loops: B {(i, i) : i N } =. We say that the graph is undirected, if (i, j) B : (j, i) B. We say that the graph is directed, if (i, j) B : (j, i) B. Path (of length k > 0): sequence of nodes n1,...,n k+1 suchthat i = 1,...,k : (n i, n i+1 ) B. Cycle: a path n1,...,n k+1 suchthat n 1 = n k+1. Connected graph: path from any node to any other node. Acyclic graph: no path is a cycle. NB: we can direct an undirected graph, and we can undirect a directed graph... Louis Wehenkel GBIO Bioinformatique (4/16)

5 Mathematical notion of tree General definition of a tree: A tree is a connected, acyclic, undirected graph. Terminology: N: number of nodes, vertices (noeuds, sommets) B: number of undirected branches, edges (arcs, arêtes) d(n): degree of node n (number of branches which are connected to n). General properties: B = N 1, n d(n) = 2B. Trivial tree: 1 node, 0 branches. Particular types of trees: Unrooted tree. Rooted tree: one of the nodes is chosen as the root; this allows to orient arcs starting with the root. Bifurcating tree Binary (rooted tree): each node has either 2 or 0 successors. Louis Wehenkel GBIO Bioinformatique (5/16)

6 Bifurcating trees Trees have two types of nodes: The external (or terminal) nodes: they have degree 1 The internal nodes (i.e. non terminal nodes): they all have degree > 1. By definition, a tree is a bifurcating tree whose internal nodes all have a degree of 3. Relation among numbers of external (E) and internal (I) nodes and branches I = E 2 ( E 3) B = 2E 3 ( E 3) Louis Wehenkel GBIO Bioinformatique (6/16)

7 Binary rooted trees Binary rooted trees have two types of nodes: The terminal (or external) nodes: they have degree 1 The internal nodes (i.e. non terminal nodes): they all have degree 3, except for the root which has degree 2. Relation among the number of teminal and internal nodes: We have: n d(n) = 2 + 3(I 1) + T for binary trees We also have: N = T + I for binary trees In general, for trees we have: B = (N 1) and n d(n) = 2B. I = T 1 and B = 2T 2 for binary trees. Louis Wehenkel GBIO Bioinformatique (7/16)

8 Illustration: undirected graphs and trees Louis Wehenkel GBIO Bioinformatique (8/16)

9 Phylogenetic trees Representation of a set of species by a binary rooted tree, where each living species corresponds to a terminal node, internal nodes correspond to ancestral species, the root node corresponds to the common ancestor, arc lengths are proportional to time that has elapsed since father. Representation of a set of genomic sequences of mutants of a virus by a binary rooted tree, each node corresponds to a homogenous population of viruses, root node corresponds to the ancestral population, the terminal nodes correspond to the currently living variants, or those that have been observed at some time spot, arc length is proportional to number of mutations (or time). NB: in some cases, we actually would need to consider non-binary trees, or even graphs that are not trees. Louis Wehenkel GBIO Bioinformatique (9/16)

10 Illustration: mitochondrial DNA of humans etc. Louis Wehenkel GBIO Bioinformatique (10/16)

11 Illustration: tree of life (from Wikipedia). Louis Wehenkel GBIO Bioinformatique (11/16)

12 We will describe the neighbor joing algorithm for inferring a binary phylogenetic tree from a distance matrix. Given a set of taxa T = {τ1,..., τ n }, and matrix of pairwise distances D i,j = d(τ i, τ j ), first build an unrooted bifurcating tree, then choose a branch and insert root node into the branch. Bifurcating tree (properties) each external node corresponds to one of the given taxa (there are T = n external nodes) each internal node has three neighbors (there are I = n 2 internal nodes) each branch has a certain length (there are B = I + T 1 = 2n 3 branches) arc lengths Li,j are computed such that d(τ i, τ j ) is well approximated by the length of (the single path) from external node i to external node j. Louis Wehenkel GBIO Bioinformatique (12/16)

13 Examples (base cases) Suppose n = 2, i.e. T = {τ 1,τ 2 }: There exists only one unrooted bifurcating tree: τ1 τ 2 Arc length = Lτ1,τ 2 = d(τ 1, τ 2 ) Root will be placed somewhere on this arc, yeilding the binary rooted tree τ 1 R τ 2. Branch lengths of rooted tree such that : L R,τ1 + L R,τ2 = d(τ 1, τ 2 ). Distances are exactly represented by the tree. Suppose n = 3, i.e. T = {τ 1,τ 2,τ 3 }: There exists only one unrooted bifurcating tree: τi A τ j τ k If branch lengths satisfy LA,τi + L A,τj = d(τ i, τ j ) the tree represents the distance matrix exactly. They can be computed by LA,τk = d(τi,τ k)+d(τ j,τ k ) d(τ i,τ j) 2, where τ k T, {τ i, τ j } = T \ {τ k }. The root is chosen arbitrarilly to split one of the three branches, A τ k. Louis Wehenkel GBIO Bioinformatique (13/16)

14 Generic greedy bottom up merging algorithm (for n 4) Initialize: N = {τ1,...,τ n }, a list of nodes to be inserted in the tree. B =, a list of branches representing the tree (a branch i j is described by a tuple (i, j, L i,j )). Node count: k = n. Iteratively shrink N and grow B, until N = : 1. If there are only two nodes i, j left in N, remove them and insert a branch (i, j, d(i, j)) into B. 2. Set node count k = k Identify nodes i, j N, and create an internal node k that will join them in the final tree. 4. Remove i and j from N; insert k in N. 5. Compute arc lengths L k,i and L k,j ; store branches (k, i, L k,i ) and (k, j, L k,j ) in B. 6. Compute distances d(k, i), i N. Choose root on one of the branches of the bifurcating tree. Louis Wehenkel GBIO Bioinformatique (14/16)

15 Neighbor joining algorithm (NJ) NJ: particular case of the above greedy algorithm: Steps 3 and 4: choice of nodes of N to merge: Compute for each node i N the number R i = P j N d(i,j) (the sum of distances to all other nodes (NB: d(i,i) = 0, i)). Compute for each pair of nodes i j N the neighborliness M i,j = ( N 2)d(i, j) R i R j. Replace pair of nodes i,j arg max i j M i,j by node k in N. Step 5: Compute arc lengths between node k and nodes i and j by (NB: N has just been decremented by 1): L(i,k) = d(i,j ) + R i R j ; L(j,k) = d(i,j ) + R j R i. 2 2( N 1) 2 2( N 1) Step 6: compute distances of new node k created by merging i, j by: d(k,l) = d(i,l)+d(j,l) d(i,j), l N \ {i,j }; d(k,k) = 0. 2 Properties of NJ: If B that represents the original distances exactly, NBJ will find one. Read article Naruya Saitou and Masatoshi Nei, 1987 for explanations and illustrations. Louis Wehenkel GBIO Bioinformatique (15/16)

16 Deadline November 27. Read chapter 7 and reference paper Naruya Saitou and Masatoshi Nei, Apply NJ for a case of 3 taxa only. Show that it indeed yields results consistent with the second basecase of slide 12. Apply NJ to the case with 5 taxa with their distances derived from the tree of Fig. 7.5 as in the table of page 117. Redo this exercise, after incrementing by 3 the distance between the taxa 1 and 2. In both cases, check whether the arc lengths of any path between two taxa sum up to the distance among these taxa. Discuss the results. Louis Wehenkel GBIO Bioinformatique (16/16)