can mathematicians find the woods?

Transcription

1 Eolutionary trees, coalescents, and gene trees: can mathematicians find the woods? Joe Felsenstein Department of Genome Sciences and Department of Biology Eolutionary trees, coalescents, and gene trees: p.1/51

2 Sketch in Darwin s notebook, 1842 Eolutionary trees, coalescents, and gene trees: p.2/51

3 The only figure in Darwin s book, 1859 Eolutionary trees, coalescents, and gene trees: p.3/51

4 Ernst Haeckel s tree, 1866 Eolutionary trees, coalescents, and gene trees: p.4/51

5 Simpson s tree of horses, 1940 s Eolutionary trees, coalescents, and gene trees: p.5/51

6 Eolution of characters along trees x 1 1 x 6 x x 3 3 x 4 4 x 8 x 5 5 x x 9 8 Characters are modeled (in simple cases) as changing by Marko processes along branches of the tree, according to graphical models like this. Eolutionary trees, coalescents, and gene trees: p.6/51

7 Eolution of characters along trees xx xxx xx xxx xx xxx xx xxx xx xxx xx xxx x 6 x 776 x 6 x 76 x xx xxx xx xxx With multiple characters (such as sites in DNA) sites are often assumed to change independently, so we hae stacked graphical models. Eolutionary trees, coalescents, and gene trees: p.7/51

8 A data example: mitochondrial D-loop sequences Eolutionary trees, coalescents, and gene trees: p.8/51

9 which gies the ML tree Chimp Human Orang Gorilla Gibbon ln L = Maximum likelihood tree for the Hasegawa 232-site mitochondrial D-loop data set, with Ts/Tn set to 2, analyzed with maximum likelihood (DNAML) Mouse Boine Eolutionary trees, coalescents, and gene trees: p.9/51

10 How many trees? Rooted, bifurcating, tips labelled species number of trees , , ,027, ,459, ,729, ,749,310, ,234,143, ,905,853,580, ,458,046,676, ,190,283,353,629, ,898,783,962,510, ,332,659,870,762,850, ,643,095,476,699,771, ,200,794,532,637,891,559, Eolutionary trees, coalescents, and gene trees: p.10/51

11 Tree space in a small case an example: three species with a clock A B C trifurcation t 1 t 2 t 1 not possible OK etc. t 2 when we consider all three possible topologies, the space looks like: t 1 t 2 Eolutionary trees, coalescents, and gene trees: p.11/51

12 Through the looking glass B A 1 C wall wall F D floor 9 E Eolutionary trees, coalescents, and gene trees: p.12/51

13 Nearest-neighbor interchanges B A 1 F C D E Eolutionary trees, coalescents, and gene trees: p.13/51

14 Nearest-neighbor interchanges B A 1 F C D E A 1 F 6 B D C E Eolutionary trees, coalescents, and gene trees: p.14/51

15 Nearest-neighbor interchanges B A 1 F C D A 1 F 6 7 B 2 3 C 8 4 D 9 5 E A 1 F 6 B E 5 4 D C E Eolutionary trees, coalescents, and gene trees: p.15/51

16 Nearest-neighbor interchanges B A 1 F C D E A 1 F 6 B E 5 4 D C A B C E F 6 4 D Eolutionary trees, coalescents, and gene trees: p.16/51

17 Nearest-neighbor interchanges B A 1 F C D A 1 F 6 7 B 2 3 C 8 4 D 9 5 E A 1 F 6 B E 5 4 D C A 1 F B C E E D Eolutionary trees, coalescents, and gene trees: p.17/51

18 The Schoenberg graph A C D B E A D B E C B C E D A A E B C D B A C D E B D C A E A B C D E A B C E D A B D C E A D B C E A C B D E C A B D E A E B D C A E C B D D A B E C Eolutionary trees, coalescents, and gene trees: p.18/51

19 Where does the true tree come from? (10 billion or so species) Model of random branching (with extinction too) Model of sampling of species by the biologist Models, Trees, and Data A E B DC model of eolution of characters along the tree Aligned DNA (or protein) sequences other sequences too A B C D E ggtca aactt gagtc aacat... ggtcc aagtt gagtc gacat... ggtca aactt gagtc aacat... ggtcc aactt gactc aatat... gctca aagtt gactc ttcat... Eolutionary trees, coalescents, and gene trees: p.19/51

20 Birth-and-death processes as priors? If they hae parameters (λ, µ), what are the priors on their alues? Eolutionary trees, coalescents, and gene trees: p.20/51

21 Birth-and-death processes as priors? If they hae parameters (λ, µ), what are the priors on their alues? How do we determine how long the BD process runs? Eolutionary trees, coalescents, and gene trees: p.20/51

22 Birth-and-death processes as priors? If they hae parameters (λ, µ), what are the priors on their alues? How do we determine how long the BD process runs? If the systematist selects species, how do we characterize that process? Eolutionary trees, coalescents, and gene trees: p.20/51

23 Birth-and-death processes as priors? If they hae parameters (λ, µ), what are the priors on their alues? How do we determine how long the BD process runs? If the systematist selects species, how do we characterize that process? How do we sample N species out of a huge tree-of-life? It s not so easy Eolutionary trees, coalescents, and gene trees: p.20/51

24 What has mathematics done for us? The analogy between recursion and trees Discoered many times, basis for dynamic programming algorithms for computing likelihoods and other similar quantities. Hadamard transform methods Hendy and Penny (1989) discoered that a Hadamard transform applied to a list of frequencies of occurrences of different patterns of character alues (at the tips of the tree) yield support for different splits in the tree. Inariants Caender and also Lake (both in 1987) initiated the exploration, not of tree space but of polynomial relationships between expected frequencies of occurrence of patterns of character alues. Related to the practice of algebraic statistics. Eolutionary trees, coalescents, and gene trees: p.21/51

25 The Hadamard Conjugation C D E B A 0.6 Tree 0.8 F 0 31 Partitions Hadamard Transform Conjugate Spectrum Eolutionary trees, coalescents, and gene trees: p.22/51

26 My ancestor? Charles the Great (Charlemagne) born 747 about 44 more generations Cornelia John Maud William Itzhak Jacob Helen Will Sheimdel Le Eleanor Jake Joe 1850s 1880s 1910s 1942 Eolutionary trees, coalescents, and gene trees: p.23/51

27 Chromosome 1, back up one lineage 6 (none) now Eolutionary trees, coalescents, and gene trees: p.24/51

28 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.25/51

29 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.26/51

30 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.27/51

31 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.28/51

32 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.29/51

33 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.30/51

34 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.31/51

35 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.32/51

36 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.33/51

37 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.34/51

38 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.35/51

39 Coalescent genealogy for one gene Time Eolutionary trees, coalescents, and gene trees: p.36/51

40 Untangling the crossed lines... Time Eolutionary trees, coalescents, and gene trees: p.37/51

41 Genealogy of a sample of 3 copies Time Eolutionary trees, coalescents, and gene trees: p.38/51

42 J. F. C. Kingman s (1982) coalescent 1. start with n tips 2. go back an amount of time( drawn from Exponential 3. join a random pair of the n 4. n n 1 5. if n = 1 stop, else go to step 2. ) 4N n(n 1) This excellently approximates the distribution of genealogies which arise from samples from a standard (Wright-Fisher) population genetics model with a population size of N, proided n 2 N Eolutionary trees, coalescents, and gene trees: p.39/51

43 A coalescent with migration among populations population 1 population 2 population 3 Eolutionary trees, coalescents, and gene trees: p.40/51

44 A coalescent with recombination Recomb. Different markers hae slightly different coalescent trees Eolutionary trees, coalescents, and gene trees: p.41/51

45 Coalescents for two genes locus A locus B both = recombination Eolutionary trees, coalescents, and gene trees: p.42/51

46 Species trees and trees of gene copies N 1 N 2 t 1 N 4 N 3 t 2 N 5 Eolutionary trees, coalescents, and gene trees: p.43/51

47 Species trees and trees of gene copies N 1 N 2 t 1 N 4 N 3 t 2 N 5 Eolutionary trees, coalescents, and gene trees: p.44/51

48 Species trees and trees of gene copies N 1 N 2 t 1 N 4 N 3 t 2 N 5 Eolutionary trees, coalescents, and gene trees: p.45/51

49 A gene duplication in a phylogeny Frog Human Monkey Squirrel a b a b a b species boundary gene duplication tree of genes Eolutionary trees, coalescents, and gene trees: p.46/51

50 If we just examine the tree of genes Frog Human Monkey Squirrel Human Monkey Squirrel a a a b b b These two trees should be identical Eolutionary trees, coalescents, and gene trees: p.47/51

51 A tree of hemoglobins (Morris Goodman, 1975) Eolutionary trees, coalescents, and gene trees: p.48/51

52 Highlighting the duplication eents Eolutionary trees, coalescents, and gene trees: p.49/51

53 References Billera, L. J., S. P. Holmes, and K. Vogtmann Geometry of the space of phylogenetic trees. Adances in Applied Mathematics 27: Chang, J. T Recent common ancestors of all present-day indiiduals. Adances in Applied Probability 31(4): Edwards, A. W. F. and L. L. Caalli-Sforza Reconstruction of eolutionary trees. pp in Phenetic and Phylogenetic Classification, ed. V. H. Heywood and J. McNeill. Systematics Association Publication No. 6. Systematics Association, London. [First paper on numerical methods for estimating phylogenies (from gene frequencies)] Felsenstein, J Inferring Phylogenies. Sinauer Associates, Sunderland, Massachusetts. [Book you and all your friends must rush out and buy] Semple, C. and M. A. Steel. Phylogenetics. Oxford Lecture Series in Mathematics and Its Applications. Oxford Uniersity Press, Oxford. Eolutionary trees, coalescents, and gene trees: p.50/51

54 How it was done This projection produced as a PDF and iewed using the Full Screen mode (in the View menu) of Adobe Acrobat Reader: I made my PDF using LaTeX (though Adobe Acrobat is another possibility): using the prosper style in LaTeX, using Latex to make a.di file, using dips to turn this into a Postscript file, using ps2pdf to make it into a PDF file, and displaying the slides in Adobe Acrobat Reader. Result: nice slides using freeware. Eolutionary trees, coalescents, and gene trees: p.51/51