Distance-based Phylogenetic Methods Near a Polytomy

Transcription

1 Distance-based Phylogenetic Methods Near a Polytomy Ruth Davidson and Seth Sullivant NCSU UIUC May 21, 2014

2 2 Phylogenetic trees model the common evolutionary history of a group of species Leaves = extant species from which we gather data (such as DNA sequences) Distance-based methods are one type of phylogenetic inference : Internal nodes = ancestral species inferred from data Least-Squares Phylogeny (LSP): NP-Hard UPGMA algorithm: O(n 2 ) Neighbor-Joining (NJ) algorithm: O(n 3 ) UPGMA and NJ are fundamental examples of distance-based algorithms and heuristics for LSP

3 3 Why study distance-based methods? Understand how well distance-based algorithms perform as polynomial-time heuristics for hard optimization problems Mathematical properties of input and output spaces give principled insight for design of phylogenetic methods Components of "divide-and-conquer" meta-methods for phylogeny estimation Associated to interesting geometry problems

4 4 Distance-based methods: inputs D vs. outputs d A dissimilarity map D on [n] = {1, 2,..., n} satisfies D(x, y) = D(y, x) for all {x, y} [n], and D(x, x) = 0. Pairwise path distances in a weighted tree induce a tree metric d on the leaves. D = 1, 2 1, 3 2, 3 (1, 3, 5) UPGMA 1, 2 1, 3 2, 3 d = (1, 4, 4) A distance-based method induces a partition of R (n 2) indexed by the combinatorial tree realizing the output of method. UPGMA and NJ send D to d according to linear inequalities, inducing a polyhedral subdivision.

5 UPGMA algorithm (Sokal and Michener) Step 1: Choose i, j minimizing D i,j, identify {i, j} as a single node, update distances D(1) given by D(1) (i,j),k = 1 2 (D i,k+d j,k ), D(1) l,m = D l,m if {l, m} {i, j} = D(1) (1,2),3 = 1 2 (D 1,3 + D 2,3 ) = 1 (3 + 5) = 4 2 D = 1, 2 1, 3 2, 3 (1, 3, 5) D(1) = (1, 2), 3 4 Step k: Choose A, B [n] minimizing D(k 1) A,B, identify A B as a single node, update distances D(k) given by D(k) (A B),C = A A + B D(k 1) A,C + B A + B D(k 1) B,C

6 6 D = UPGMA inputs and outputs in R (3 2) 1, 2 1, 3 2, 3 (1, 3, 5) 0 1, 2 1, 3 2, 3 d = (1, 4, 4) Polyhedral fan of equidistant tree metrics on [3] R (3 2) 0.

7 7 Distance-based algorithms are a heuristic for LSP Problem (Least-Squares Phylogeny). Given D R (n 2), minimize D d 2 subject to d a tree metric. Theorem (Day, 1987). LSP is NP-Hard. Theorem (Atteson, 1999). Let d be a tree metric such that the shortest edge length in a binary tree representation of d is w > 0. Suppose D R (n 2) satisfies D d < w/2. Then NJ (D) = ˆd, where ˆd and d are realized by trees with the same shape. What if we allow an edge weight to be zero? In a neighborhood of a tree metric with a polytomy, or unresolved vertex, will a distance-based algorithm choose the best tree shape?

8 Tritomy Tree Metrics A tritomy tree metric is a point at the intersection of 3 maximal cones in the fan of tree metrics with three binary resolutions. For n > 3, tritomies live at intersection of 3 maximal cones which share a common set K of extreme rays and each possess one unique ray r. Projections of the unique rays onto Span(K ) form a 1-dimensional fan spanning a 2-dimensional plane. LSP region ratios in the plane LSP region ratios in R (n 2) 0.

9 Formulae for Angles Between Tritomy Tree Cones Let a = A, b = B, c = C, and d = D. Theorem (D -Sullivant). The angle between the AC B tree and the ( BC A tree is ) c arccos. (a + c)(b + c) Theorem (D-Sullivant). The angle between the AC BD and AD BC trees is ( ) ab + cd arccos. (a + c)(b + c)(a + d)(b + d)

10 10 Geometry of the input space near a tritomy Theorem (D-Sullivant). The geometry of UPGMA (red) and LSP regions (blue) near a tritomy depends only on subtree sizes. UPGMA poorly approximates LSP in some cases. c a b a b c a b c d. LSP boundaries = angle bisectors. UPGMA boundaries = reflected tree rays.

11 1 Simulations show NJ regions depend on tree shape NJ and LSP near d 1 and d 2 Resolution LSP NJ : d 1 NJ: d 2 AB CD AC BD AD BC Percentages generated by running NJ on points sampled uniformly on small spheres around d 1 and d 2.

12 12 Summary and future work The polyhedral geometry of LSP and UPGMA regions near a tritomy depends only on the subclade sizes for the tritomy. Near tritomies, UPGMA and NJ often poorly match LSP. Size of NJ regions near a tritomy depends on the branching structure of the associated tree. What is the geometry of LSP, UPGMA, and NJ regions near higher-degree polytomies? Study decomposition of R (n 2) induced by BIONJ.

13 13 Thanks! www4.ncsu.edu/ redavids K. Atteson. The performance of the NJ method of phylogeny reconstruction. Algorithmica 25 (1999): W.M. Bruno, N.D. Socci, and A.L. Halpern. Weighted Neighbor Joining: a likelihood-based approach to distance- based phylogeny reconstruction. Molecular Biology and Evolution 17 (2000): R. Davidson and S. Sullivant. Distance-based phylogenetic methods around a polytomy. To appear in IEEE/ACM Transactions on Computational Biology and Bioinformatics arxiv: R. Davidson and S. Sullivant. Polyhedral combinatorics of UPGMA cones. Advances in Applied Mathematics 50 (2013), no. 2: W. Day. Computational complexity of inferring phylogenies from dissimilarity matrices. Bulletin of Mathematical Biology. 49 (1987): O. Gascuel. BIONJ: an improved version of the NJ algorithm based on a simple model of sequence data. Molecular Biology and Evolution 14 (1997): N. Saitou and M. Nei. The neighbor joining method: a new method for reconstructing phylogenetic trees. Molecular Biology and Evolution 4 (1987), no. 4: R.R. Sokal and C. Michener. A statistical method for evaluating systematic relationships. University of Kansas Science Bulletin 38 (1958):