Phylogenetic trees Sushmita Roy BMI/CS 576 www.biostat.wisc.edu/bmi576/ sroy@biostat.wisc.edu Oct 1st, 2013

Key concepts in this section What are phylogenies or phylogenetic trees? Terminology such as extant, ancestral, branch point, branch length, orthologs, paralogs, taxon Why build phylogenetic trees? How to build phylogenetic trees? Distance-based methods Parsimony methods Minimize the number of changes Probabilistic methods Find the tree that best explains the data using probabilistic models

What are phylogenetic trees? A tree that describes evolutionary relationships among entities Species, genes, strains Leaves represent extant entities Internal nodes represent ancestral species Such a tree is inferred from observations in existing organisms.

Tree of life aims to represents the phylogeny of all species on earth From http://tellapallet.com/tree_of_life.htm

Phylogenetic tree of 29 mammals Lindbald-Toh et al, 2011, Nature

Why phylogenetic trees? Understand how organisms are related Do humans and chimpanzees share a common ancestor or do humans and gorillas? Ask how closely organisms are related Humans and chimpanzees shard a common ancestor 5mya Provide insight into the evolutionary history of species How specific functions have evolved Language evolution Identify signatures of conservation of sequence Conjecture the fate of specific regions of the genome Will the human Y disappear? Inform multiple sequence alignments

Orthologs and paralogs Two sequences in two species that have a a common ancestor Diverged due to a speciation event Used to create a “species tree” Paralogs: Two sequences in the same species that arose from a gene duplication event Captured in a “gene tree”.

Phylogenetic tree basics Leaves represent things (genes, species, individuals/strains) being compared the term taxon (taxa plural) is used to refer to these when they represent species and broader classifications of organisms For example if taxa are species, the tree is a species tree Internal nodes are hypothetical ancestral units Phylogenetic trees can be rooted or unrooted the root represents the common ancestor In a rooted tree, path from root to a node represents an evolutionary path Gives directionality to evolutionary time An unrooted tree specifies relationships among taxa, but not from an ancestor

Tree basics Internal node: Ancestral 2 5 9 8 8 6 Branch length 6 7 7 1 4 1 2 3 4 5 3 Leaf node: Extant Unrooted tree Rooted tree Each tree topology represents a different evolutionary history For a species tree, internal nodes represent speciation events

Internal nodes represent ancestral species Tree of Life project (http://tolweb.org/tree/)

Rooting a tree An unrooted tree can be converted to a rooted tree using an outgroup species Outgroup: a species known to be more distantly related all the species than each of the species themselves Find the branch where the outgroup is selected to be added That gives the root

Tree counting A rooted tree with n leaf nodes has n-1 internal nodes 2n-2 edges/branches An unrooted tree with n leaf nodes has n-2 internal nodes 2n-3 edges/branches A root can be added to any of these branches to give 2n-3 rooted trees for any unrooted tree For three taxa there is one unrooted tree and three rooted trees

Tree counting 1 1 3 3 1 2 3 2 2 1 An unrooted tree 3 3 2 1 2 1 3 2 1 3 Possible positions for root Rooted trees

Tree counting Instead of adding a root we could add a branch for the n+1th taxon 4 1 1 1 3 3 3 2 2 2 1 1 4 3 2 3 2 1 1 3 3 2 2 4

Tree counting With four nodes, we have five branches Each of the branches can give rise to five trees of six nodes Thus we have 3*5 trees In general for n nodes we can have (1)(3)(5)..(2n-5) unrooted trees

Constructing phylogenetic trees Three types of methods Distance based methods Parsimony methods Probabilistic approaches Most methods start with pairwise distance methods We have already seen one method!

Methods for phylogenetic tree reconstruction Distance-based methods UPGMA Neighbor joining Assume additivity and sometimes a “molecular clock” Additivity means we can add up the branch lengths of the tree connecting two nodes and get their distances. Alignment-based methods Parsimony Probabilistic

Defining distance between sequences Fractional alignment difference for two sequences i and j pij = mij/Lij Gives an estimate of changes per site mij: Number of mismatches Assumes that changes have happened only once Underestimates the distance between sequences Assumes all sequences change at the same rate Jukes Cantor distance The simplest, evolutionary distance

UPGMA relies on the molecular clock assumption Sequences diverge at the same rate at different points in the phylogeny Distance from any leaf to root is the same. If this is true the data is said to be ultrametric

The molecular clock assumption & ultrametric data Ultrametric data: for any triplet of sequences, i, j, k, the distances are either all equal, or two are equal and the remaining one is smaller A B C D E 8 5 3 4 3 d_ij <= max(d_ik, d_jk) 2 1 A E D B C

Problems with the molecular clock assumption 3 2 4 2 3 4 1 1 Constructed by UPGMA Actual tree

Neighbor joining The ultra-metric property is too strong Most sequences diverge at different rates A more relaxed requirement is that of additivity Distance between a pair of species/nodes is equal to the sum of the branch lengths Uses a similar idea to construct trees as UPGMA That is consider pairs of nodes and joins them Produces unrooted trees

How to select nodes for merging? Given all pairwise distances for n sequences dij denote the distance between node i and j Should we select node pairs with the smallest dij? A B C D 0.4 0.1 Should we merge A and B?

Need to correct for long branches L: current set of leaves ri : Average distance from other nodes

Defining the distance to a new node dkm? i m k j New node Given dij, dim, djm, how to calculate distance to new node k?

Algorithm for NJ Initialization Iteration Terminate T be set the of leaf nodes L = T Estimate ri for all i in L Estimate Dij Iteration Pick a pair i, j from L such that Dij is smallest Define new node k Estimate dik, djk, add edge between k and i, and between k to j Add k to T, remove i and j from L Estimate Dmn for all nodes m, n in L Terminate If L has two nodes, add the edge between these two.

An example with neighbor joining Consider 5 sequences: A, B, C, D, E Distance matrix What is the tree inferred by the Neighbor joining algorithm? B C D E A 5 4 9 8 10 7 6 B C D E

Can we check for additivity? Check for additivity: For four leaves, i, j, k, l and the distances dij, dik, dil, djk, djl, dkl j i l k The three sums of two distances j j i j i i l l l k k k Should be such that two of these are equal, and larger than the third.