Lecture 3. Phylogeny methods: Branch and bound, distance methods

Transcription

1 Lecture 3. Phylogeny methods: ranch and bound, distance methods Joe Felsenstein epartment of Genome Sciences and epartment of iology Lecture 3. Phylogeny methods: ranch and bound, distance methods p.1/25

2 Greedy search by sequential addition Greedy search by addition of species in a fixed order (,,,, ) in the best place each time. Lecture 3. Phylogeny methods: ranch and bound, distance methods p.2/25

3 Goloboff s time-saving trick G H K G H R L V Z S U V Z M R S U Goloboff s economy in computing scores of rearranged trees Once the views have been computed, they can be taken to represent subtrees, without going inside those subtrees Lecture 3. Phylogeny methods: ranch and bound, distance methods p.3/25

4 Star decomposition F F F F F F Star decomposition" search for best tree can happen in multiple ways Lecture 3. Phylogeny methods: ranch and bound, distance methods p.4/25

5 isk-covering F isk covering" assembly of a tree from overlapping estimated subtrees Lecture 3. Phylogeny methods: ranch and bound, distance methods p.5/25

6 Shortest Hamiltonian path problem (a) (b) (c) (d) Lecture 3. Phylogeny methods: ranch and bound, distance methods p.6/25

7 Search tree for this problem (1,2,3,4,5,6,7,8,10,9) (1,2,3,4,5,6,7,9,10,8) (1,2,3,4,5,6,7,10,9,8) (1,2,3,4,5,6,7,8,9,10) (1,2,3,4,5,6,7,9,8,10) (1,2,3,4,5,6,7,10,8,9) add add 310 add 9 add add 310 add 8 add add 39 add 8 add 9 add 10 add 8 add 10 add 8 add 9 add 8 add 10 add 9 etc. etc. add 3 add 4 add 5 etc. etc. etc. add 2 add 3 add 4 add 5 etc. etc. add 1 add 2 add 3 start Lecture 3. Phylogeny methods: ranch and bound, distance methods p.7/25

8 Search tree of trees Lecture 3. Phylogeny methods: ranch and bound, distance methods p.8/25

9 same, with parsimony scores in place of trees Lecture 3. Phylogeny methods: ranch and bound, distance methods p.9/25

10 Time Polynomial time and exponential time n +4n 3 e 0.5n Problem size How does the time taken by an algorithm depend on the size of the problem? If it is a polynomial (even one with big coefficients), with a big enough case it is faster than one that depends on the size exponentially. Lecture 3. Phylogeny methods: ranch and bound, distance methods p.10/25

11 NP completeness and NP hardness P NP does this part exist? is P = NP? NP Hard NP omplete (This diagram is not quite correct see the diagrams on the Wikipedia page for NP-hard ). P = problems that can be solved by a polynomial time algorithm NP complete = problems for which a proposed solution can be checked in polynomial time but for which it can be proven that if one of them is in P, all are. NP hard = problems for which a solution can be checked in polynomial time, but might be not solvable in polynomial time. Lecture 3. Phylogeny methods: ranch and bound, distance methods p.11/25

12 istance methods These have been attractive, particular to mathematical scientists who love geometry. This has its good and bad effects. 1. Take the sequences in all pairs. 2. For each pair compute a distance. (s we will see, this is best thought of as the length of the 2-species tree for those species). 3. Try to find that tree which best fits the table of distances. Lecture 3. Phylogeny methods: ranch and bound, distance methods p.12/25

13 phylogeny with branch lengths and the pairwise distances it predicts Lecture 3. Phylogeny methods: ranch and bound, distance methods p.13/25

14 phylogeny with branch lengths v 1 v 2 v 5 v 6 v 7 v 3 v 4 Lecture 3. Phylogeny methods: ranch and bound, distance methods p.14/25

15 Least squares trees Least squares methods minimize Q = n w ij ( ij d ij ) 2 i=1 j i over all trees, using the distances d ij that they predict. avalli-sforza and dwards suggested w ij = 1, Fitch and Margoliash suggested w ij = 1/ 2 ij. Lecture 3. Phylogeny methods: ranch and bound, distance methods p.15/25

16 Statistical assumptions of least squares trees Implicit assumption is that distances are (independently?) Normally distributed with expectation d ij and variance proportional to 1/w 2 ij : ij N (d ij, K/w ij ) Thus the different weightings correspond to different assumptions about the error in the distances. lso, there is assumed to be no covariance of distances. In fact, the distances will covary, since a change in an interior branch of the tree increases (or decreases) all distances whose paths go through that branch. Lecture 3. Phylogeny methods: ranch and bound, distance methods p.16/25

17 Matrix approach to fitting branch lengths If we stack the distances up into a column vector, we can solve the least squares equation (obtained by taking derivatives of the quadratic form Q): T = ( 12, 13, 14, 15, 23, 24, 25, 34, 35, 45 ) X T = X T X v. where the design matrix X for the given tree topology has 1 s whenever a given branch lies on the path between those two species. Here is the design matrix for the tree we just saw. X = ranches which , , , , , , , , , ,5 v v 2 1 v 7 v 5 v 6 v v 3 4 Lecture 3. Phylogeny methods: ranch and bound, distance methods p.17/25

18 The Jukes-antor model for N u/3 G u/3 u/3 u/3 u/3 u/3 T Lecture 3. Phylogeny methods: ranch and bound, distance methods p.18/25

19 erivation of the probability of change 1. Imagine events occuring at rate 4 3u per unit time which replace a base by one of the 4 bases chosen at random. 2. Persuade yourself that this is no different in outcome from events u per unit time that replace it by one of the other 3 chosen at random. 3. The probability a branch has none of these (first kind of) events if it is of length t is exp( 4 3 u t). (Think the zero term of a Poisson distribution). 4. If it does have one or more of these events, you end up with one of the 4 bases chosen at random. 5. Therefore the probability of a net change is: 3 4 (1 e ( 4 3 u t)) Lecture 3. Phylogeny methods: ranch and bound, distance methods p.19/25

20 per site differences The distance for the Jukes-antor model branch length Lecture 3. Phylogeny methods: ranch and bound, distance methods p.20/25

21 If you don t correct for multiple hits Left: the true tree. Right: a tree fitting the uncorrected distances Lecture 3. Phylogeny methods: ranch and bound, distance methods p.21/25

22 References, page 1 Maddison,. R The discovery and importance of multiple islands of most-parsimonious trees. Systematic Zoology 40: [iscusses heuristic search strategy involving ties, multiple starts] Farris, J. S Methods for computing Wagner trees. Systematic Zoology 19: [arly parsimony algorithms paper is one of first to mention sequential addition strategy] Saitou, N., and M. Nei The neighbor-joining method: a new method for reconstructing phylogenetic trees. Molecular iology and volution 4: [First mention of star-decomposition search for best trees, sort of] Strimmer, K., and. von Haeseler Quartet puzzling: a quartet maximum likelihood method for reconstructing tree topologies. Molecular iology and volution 13: [ssembles trees out of quartets] Huson,., S. Nettles, L. Parida, T. Warnow, and S. Yooseph The disk-covering method for tree reconstruction. pp in Proceedings of lgorithms and xperiments (LX98), Trento, Italy, Feb. 9-11, 1998, ed. R. attiti and.. ertossi. [ isk-covering method for long stringy trees] Lecture 3. Phylogeny methods: ranch and bound, distance methods p.22/25

23 References, page 2 Foulds, L. R. and R. L. Graham The Steiner problem in phylogeny is NP-complete. dvances in pplied Mathematics 3: [Parsimony is NP-hard] Graham, R. L. and L. R. Foulds Unlikelihood that minimal phylogenies for a realistic biological study can be constructed in reasonable computat ional time. Mathematical iosciences 60: [... and more] Hendy, M.. and. Penny ranch and bound algorithms to determine minimal evolutionary trees. Mathematical iosciences 60: [Introduced branch-and-bound for phylogenies] Felsenstein, J Inferring Phylogenies. Sinauer ssociates, Sunderland, Massachusetts. [For this lecture the material is chapters 4, and 5] Semple,. and M. Steel Phylogenetics. Oxford University Press, Oxford. [lso covers search strategies] Lecture 3. Phylogeny methods: ranch and bound, distance methods p.23/25

24 References, page 3 Felsenstein, J istance methods for inferring phylogenies: a justification. volution 38: [rgument for statistical interpretation of distance methods] Farris, J. S istance data revisited. ladistics 1: [Reply to my 1984 paper] Felsenstein, J istance methods: reply to Farris. ladistics 2: [reply to Farris 1985] Farris, J. S istances and statistics. ladistics 2: [debate was cut off after this] Lecture 3. Phylogeny methods: ranch and bound, distance methods p.24/25

25 References, page 4 ryant,., and P. Waddell Rapid evaluation of least-squares and minimum-evolution criteria on phylogenetic trees. Molecular iology and volution 15: [quicker least squares distance trees] Felsenstein, J Inferring Phylogenies. Sinauer ssociates, Sunderland, Massachusetts. [See chapter 11] Semple,. and M. Steel Phylogenetics. Oxford University Press, Oxford. [See pp ] Yang, Z omputational Molecular volution. Oxford University Press, Oxford. [See pages 89-93] Lecture 3. Phylogeny methods: ranch and bound, distance methods p.25/25