1. Distance based phylogenetics
Usman Roshan

2. 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

3. 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…

4. Phylogeny Problem U V W X Y X U Y V W AGGGCAT TAGCCCA TAGACTT TGCACAA
TGCGCTT X U
say hello Y V W

5. Bipartitions
Phylogenies are equivalent to bipartitions

6. Topological differences

7. 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

8. 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
Accurate but computationally very hard

9. Distance-based methods

10. 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

11. 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)

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

13. 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

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

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

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

17. 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)

18. 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

19. Evolutionary distance estimation

20. 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

21. 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

22. 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

23. 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

24. Distance methods

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

26. Additivity

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

28. UPGMA
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

29. UPGMA A B C D A 6 26 26 13 13 B 26 26 C 6 3 3 3 3 D A B C D

30. UPGMA Doesn’t work (in general) for non-ultrametric trees A B C D 3 313 16 26 3 3 B 12 19 10 B C 10 C 13 D A D

31. UPGMA UPGMA constructs incorrect tree here 7.25 A B C D 7.25 A 13 1626
7.25
7.25 B 12 19 6 6 C 13 B C A D D

32. UPGMA Bipartition (BC,AD) is not in true tree 7.25 3 3 3 3 7.25 10 106 6 A D B C A D
True tree
UPGMA tree

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

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

35. NJ NJ constructs the correct tree for additive matrices A B C D 3 3 A13 16 26 3 3 B 12 19 10 B C 10 C 13 D A D

36. Simulation studies

37. 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

38. 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

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

40. Sequence length requirements

41. Sequence length requirements

42. Typical performance study

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

44. Error rate of NJ