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A talk given at the Algorithms for Threat Detection Program Review, March 10th, 2014.

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- 1. Quantifying MCMC exploration of phylogenetic tree space Christopher Whidden and Frederick “Erick” A. Matsen IV Fred Hutchinson Cancer Research Center http://matsen.fhcrc.org @ematsen
- 2. Phylogenetics: reconstruct evolutionary history from DNA armadillo DNA or RNA sequence data "phylogenetics" human rat giraﬀe
- 3. Phylogenetics helps us learn how HIV-1 came to be Etienne, Hahn, Sharp, Matsen and Emerman, Cell Host & Microbe, 2013
- 4. We are fond of statistical approaches to phylogenetics These are important when one would like a clear notion of uncertainty (like medicine, epidemiology, and biodefense!)
- 5. We are fond of statistical approaches to phylogenetics In particular, Bayesian methods fall into this category and have become quite popular. ACATGGCTC... ATACGTTCC... TTACGGTTC... ATCCGGTAC... ATACAGTCT... ... We can’t solve for this posterior distribution, but we can satisfy our needs by getting a big sample from it.
- 6. Markov chain Monte Carlo (MCMC) Metropolis et al., 1953. Set up a simulation such that the amount of time spent in a given state is proportional to the posterior probability of that state.
- 7. Here we want a posterior on trees If we want to use the same strategy to get a posterior on phylogenetic trees. . . ACATGGCTC... ATACGTTCC... TTACGGTTC... ATCCGGTAC... ATACAGTCT... ... we need a way to move from one phylogenetic tree to another.
- 8. Subtree-prune-regraft (SPR) deﬁnition 1 2 3 4 5 6 1 4 5 6 2 3 1 4 5 2 3 6
- 9. The set of trees as a graph connected by SPR moves (Figure from Mossel and Vigoda, Science, 2005).
- 10. This graph is connected, and every tree has nonzero posterior probability, so MCMC works† We are guaranteed to converge to the posterior distribution on trees by using Metropolis-Hastings moves built on these SPRs. That is, by bouncing around “tree space” we can get a good idea of a set of good trees. † That is, it works if we run the MCMC forever
- 11. We can’t run it forever. News ﬂash: 5 million < ∞
- 12. With pathological data, can be hard to traverse peaks goodness
- 13. We wanted to know: does this happen in real data sets? Lots of discussion in literature, but few clear conclusions. In order to understand the reasons diﬀerentiating “easy” and “diﬃcult” data sets for phylogenetic MCMC, we wanted to make it possible to visualize tree space with a relevant geometry. So, what trees are close to each other in terms of SPR moves?
- 14. dSPR : how many SPR moves from one tree to another? Say T1 T2 if there is an SPR transformation of T1 to T2 . dSPR (T , S) = T1 min ··· Tk =S k This distance is NP-hard to compute. That’s no fun!
- 15. Meet Chris Whidden, algorithms strongman In a series of four very technical papers, Chris took exact computation of dSPR from O(infeasible) to O(feasible). Then he joined my group!
- 16. Let’s take some common data sets and see what we see These are completely standard data sets of the sort that biologists analyze every day: slowly evolving nuclear, mitochondrial, or chloroplast genes. Also used as examples in: Lakner et al., Syst. Biol., 2008 Hohna and Drummond, Syst. Biol., 2012 Larget, Syst. Biol., 2013
- 17. Interested in high probability subsets of the SPR graph
- 18. Summarize by subsetting to high probability nodes node size proportional to posterior probability, and color shows distance to the highest PP tree.
- 19. The top 4096 trees for a data set
- 20. The top 4096 trees for a data set What's up with this stuﬀ? Is it important? Is it diﬃcult for the MCMC to see?
- 21. Commute time deﬁnition Commute time for a node y : how long to make the round trip from y to the highest posterior probability tree and back? Any round trip path counts!
- 22. Commute time deﬁnition Commute time for a node y : how long to make the round trip from y to the highest posterior probability tree and back? Any round trip path counts!
- 23. Commute time plot for this data set
- 24. The separation is problematic indeed Yep, those parts of the posterior are important and MCMC has trouble entering them.
- 25. Trees with 95% of posterior probability for another data set
- 26. We can use our methods to identify source of bottlenecks Hyla_cinerea Hyla_cinerea Bufo_valliceps Bufo_valliceps Nesomantis_thomasseti Hypogeophis_rostratus Eleutherodactylus_cuneatus Grandisonia_alternans Gastrophryne_carolinensis Amphiuma_tridactylum Hypogeophis_rostratus Ichthyophis_bannanicus Grandisonia_alternans Ambystoma_mexicanum Amphiuma_tridactylum Siren_intermedia Ichthyophis_bannanicus Typhlonectes_natans Plethodon_yonhalossee Discoglossus_pictus Scaphiopus_holbrooki Plethodon_yonhalossee Discoglossus_pictus Scaphiopus_holbrooki Ambystoma_mexicanum Nesomantis_thomasseti Siren_intermedia Eleutherodactylus_cuneatus Typhlonectes_natans Gastrophryne_carolinensis Xenopus_laevis Xenopus_laevis Homo_sapiens Homo_sapiens Mus_musculus Mus_musculus Rattus_norvegicus Rattus_norvegicus Oryctolagus_cuniculus Oryctolagus_cuniculus Turdus_migratorius Turdus_migratorius Gallus_gallus Gallus_gallus Heterodon_platyrhinos Heterodon_platyrhinos Sceloporus_undulatus Sceloporus_undulatus Alligator_mississippiensis Alligator_mississippiensis Trachemys_scripta Trachemys_scripta Latimeria_chalumnae Latimeria_chalumnae These are the trees at the two peaks of the connected components. Indeed, it’s very tricky to get between them!
- 27. Multidimensional scaling visualizations via dSPR
- 28. In general, a new way to explore tree space
- 29. Our applications: it’s party time Automatic identiﬁcation of (multiple) peaks in posteriors Performance of Metropolis-coupled Markov chain Monte Carlo for getting between peaks Accuracy of new “mean-ﬁeld” posterior probability approximations The ﬁrst topological convergence diagnostic These empirical investigations set the stage for additional theoretical development, and suggest new ways to move around tree space. This will translate into better phylogenetic uncertainty estimates, and hence better preparedness and response to biological threats.
- 30. Thank you Robert Beiko (Dalhousie University) Aaron Darling (University of Technology, Sydney) Connor McCoy (Fred Hutchinson Cancer Research Center) NSF award 1223057

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