Distance based phylogenetics Usman Roshan Phylogenetics Study of

Distance based phylogenetics Usman Roshan

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

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…

Phylogeny Problem U AGGGCAT V W TAGCCCA X TAGACTT Y TGCACAA X U Y V W TGCGCTT

Bipartitions • Phylogenies are equivalent to bipartitions

Topological differences

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

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

Distance-based methods

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

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)

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

Additivity • Assume we have a path of k edges and that p 1, p 2, …, 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

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

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

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

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)

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

Evolutionary distance estimation

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

Estimating evolutionary distances • Assume nucleotide changes are Bernoulli trials (i. i. d. trials of success or failure) ACCTGTGGGTAACCACCC ACCTGAGGGATAGGTCCG 00000110001001 • is probability of head in n Bernoulli trials (n is sequence length) • Compute a maximum likelihood estimate for

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

Estimating evolutionary distances • = 5/18 ACCTGTGGGTAACCACCC ACCTGAGGGATAGGTCCG 00000110001001 • Continuing in this manner we estimate all pairs of sequences in the alignment • We now have a distance matrix under a biologically sound evolutionary model for

Distance methods

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

Additivity

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

UPGMA 1. 2. 3. 4. 5. 6. 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

UPGMA A A B C D B C 6 26 26 D 10 10 26 26 6 3 A 3 3 B C 3 D

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

UPGMA constructs incorrect tree here A B C D A 13 19 26 B 12 19 C 13 D B C A D

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

Neighbor joining 1. 2. 3. 4. Additive and O(n^3) time Initialization: same as UPGMA For each species compute 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

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

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

Simulation studies

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

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

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

Sequence length requirements

Sequence length requirements

Typical performance study

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

Error rate of NJ