Neighbor-Joining with Interval Methods D. Levy 1 , Raazesh Sainudiin 2 , R. Yoshida 3 , and L. Pachter 1 1 University of California at Berkeley, Berkeley, CA 94720, USA; 2 Cornell University, Ithaca NY 14850; 3 Duke University, Durham, NC 27708, USA The software package MJOIN is available at http://bio.math.berkeley.edu/mjoin/ Introduction • The Neighbor-Joining algorithm is a recursive procedure to reconstruct a phyloge- netic tree using a transformation of pairwise distances between leaves for identifying cherries in the tree. • Pachter and Speyer showed that we can recover an n-leaf tree from the weights of m-leaf subtrees if n ≥ 2m - 1 [PS04]. • We generalized the cherry picking criterion with estimates of the weights of m-leaf subtrees. • We showed that a reconstructed tree from such weights is more accurate than one using pairwise distances. • This leads to an improved neighbor-joining algorithm whose total running time is still polynomial in the number of taxa. Neighbor Joining with Pairwise Distances Theorem. (the cherry picking criterion) [SN87, SK88] Suppose D(ij ) is a pairwise distance between taxa i and j . Then, {i, j } is a cherry if A ij = D(ij ) - (r i + r j )/(n - 2), where r i := ∑ n k =1 D(ik ), is minimal. Idea. Initialize a star-like tree and find a cherry. Then we compute branch length from the interior node to each leaf. Repeat this process recursively until we find all cherries. 1 2 3 4 8 7 6 5 2 5 4 1 2 1 5 1 2 2 5 2 1 8 8 8 8 8 7 7 7 7 7 6 6 6 6 6 5 5 5 5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1 Y X Y X Y X X Y Y X Z Figure 1: The traditional Neighbor Joining with pairwise distances. Neighbor Joining with Subtree Weights Notation. Let [n] denote the set {1, 2, ..., n} and  [n] m  denote the set of all m-element subsets of [n]. Definition. A m-dissimilarity map is a function D :  [n] m  → R ≥0 . In terms of phylogeny, this corresponds to the weights of m-subtree weights of a tree T . Theorem. Let D m be be an m-dissimilarity map on n leaves, D m :  [n] m  → R ≥0 correspond to the weights of m-subtree weights of a tree T and we define S (ij ) :=  X ∈ ( [n]\{i, j } m-2 ) D m (ijX ). Then S (ij ) is a tree metric. Furthermore, if T ′ is the additive tree corresponding to this tree metric then T ′ and T have the same tree topology and there is an invertible linear map between their edge weights. Algorithm. (Neighbor Joining with Subtree Weights) • Input: n many DNA sequences. • Output: A phylogenetic tree T with n leaves. 1. Compute all m-subtree weights via the maximum likelihood. 2. Compute S (ij ) for each pair of leaves i and j . 3. Apply Neighbor Joining method with a tree metric S (ij ) and obtain additive tree T ′ . 4.Using a linear mapping, obtain a weight of each internal edge and each leaf edge of T . Cherry Picking Theorem Theorem. Let T be a tree with n leaves and no nodes of degree 2 and let m be an integer satisfying 2 ≤ m ≤ n - 2. Let D :  [n] m  → R ≥0 be the m-dissimilarity map corresponding to the weights of the subtrees of size m in T . If Q D (ab) is a minimal element of the matrix Q D (ab)=  n - 2 m - 1   X ∈ ( [n]\{i, j } m-2 ) D(ijX ) -  X ∈ ( [n]\{i} m-1 ) D(iX ) -  X ∈ ( [n]\{j } m-1 ) D(jX ) then {a, b} is a cherry in the tree T . Note. The theorem by Saitou-Nei and Studier-Keppler is a corollary from Cherry Picking Theorem. Time Complexity If m ≥ 3, the time complexity of this algorithm is O (n m ), where n is the number of leaves of T and if m = 2, then the time complexity of this algorithm is O (n 3 ). Note: The running time complexity of the algorithm is O (n 3 ) for both m = 2 and m = 3. Interval Methods • In [LYP04], Dissimilarity maps are computed via fastDNAml which implements a gradient flow algorithm with floating-point arithmetic. • Instead, apply the rigorously enclosed maximum likelihood estimations [Sai04]. • Dissimilarity maps computed via the rigorously enclosed MLEs are guaranteed to be enclosed. Thus, reconstructed trees via the generalized NJ method with these dissimilarity maps are more accurate. Computational Results • Problem: Find the NJ tree for 21 S-locus receptor kinase (SRK) sequences [SWY + 05] involved in the self/nonself discriminating self-incompatibility system of the mustard family [Nas02]. • Result: Symmetric difference (Δ) between 10, 000 trees sampled from the likeli- hood function via MCMC and the trees reconstructed by 5 methods. DNAml was used in two ways: DNAml(A) is a basic search with no global rearrange- ments, whereas DNAml(B) applies a broader search with global rearrangements and 100 jumbled inputs. Δ NRGNJ fastDNAml DNAml(A) DNAml(B) TrExML 0 0 0 2 3608 0 2 0 0 1 471 0 4 171 6 3619 5614 0 6 5687 5 463 294 5 8 4134 3987 5636 13 71 10 8 5720 269 0 3634 12 0 272 10 0 652 14 0 10 0 0 5631 16 0 0 0 0 7 References [LYP04] D Levy, R Yoshida, and L Pachter. Neighbor joining with subtree weights. preprint, 2004. [Nas02] JB Nasrallah. Recognition and rejection of self in plant reproduction. Science, 296:305–308, 2002. [PS04] L. Pachter and D. Speyer. Reconstructing trees from subtree weights. Applied Mathematics Letters, 17:615 – 621, 2004. [Sai04] R Sainudiin. Enclosing the maximum likelihood of the simplest DNA model evolving on fixed topologies: towards a rigorous framework for phylogenetic inference. Technical Report BU1653-M, Department of Biol. Stats. and Comp. Bio., Cornell University, 2004. [SK88] J. A. Studier and K. J. Keppler. A note on the neighbor-joining method of saito and nei. Mol. Biol. Evol., 5:729 – 731, 1988. [SN87] N. Saitou and M. Nei. The neighbor joining method: a new method for reconstructing phylogenetic trees. 1987. [SWY + 05]R Sainudiin, SW Wong, K Yogeeswaran, J Nasrallah, Z Yang, and R Nielsen. Detecting site- specific physicochemical selective pressures: applications to the class-I HLA of the human major histocompatibility complex and the SRK of the plant sporophytic self-incompatibility system. Journal of Molecular Evolution, in press, 2005.