Fast Algorithms for Minimum Evolution Richard Desper, NCBI Olivier Gascuel, LIRMM

Overview I. Statement of phylogeny reconstruction problem and various approaches to solving it. II. Tree length formula as a function of average distances. III. Greedy algorithms for tree building and tree swapping. IV. Simulation results. V. A few extras regarding consistency and branch lengths.

Phylogeny Reconstruction  General problem: reconstruct the evolutionary history for a set L of extant species.  Input: multiple sequence alignment for L or matrix of estimates of pairwise evolutionary distances.  Output: weighted phylogeny representing history of L and common ancestors.

Methods  Likelihood methods : model-based likelihood maximization.  Parsimony methods : minimize total number of mutations in tree.  Distance methods: fit tree structure to inferred evolutionary distances. Leading methods include Felsenstein-Fitch-Margoliash weighted least-squares and Neighbor-Joining and its variants.

Felsenstein-Fitch-Margoliash Least-squares Method  FITCH searches the space of topologies by iteratively adding leaves and by tree swapping.  Edge weights and topology are chosen to minimize the sum of squares ( D is the input metric, D T is the induced tree metric): If s ij = 1 for all i and j, this is called the ordinary least-squares method.

Minimum Evolution  Developed by Rzhetsky and Nei (1992) as a modification of the OLS method  For each topology T, Define function l assigning OLS lengths to edges of T Define size of tree Choose T minimizing l ( T )

Recursive Definition of D A|B If A = {a}, B = {b}, D A|B = D ab, All average distances for all pairs of non-intersecting subtrees of a given topology can be calculated in O(n 2 ) time. For

External OLS Edge Length Function If e is the edge connecting the leaf i to the subtrees A and B, e B A i

Internal OLS Edge Length Function The length of the edge e is (Vach, 1988) e AC D B where

Tree length formula Lemma : with T as to the right, let denote the root of subtree X, and the edge to X for Then, C e A D B

Tree Length Formula With T as in prior slide, Using lemma and branch length formula for l(e),

General approach  To search the space of topologies, we’ll keep in memory two data structures: Sizes of each subtree of given topology Matrix of average distances  X|Y for X,Y disjoint subtrees in given topology  As we move from one topology to another, we’ll update the matrix, but only as much as needed, in an efficient manner.

Tree Swapping by NNI C e A D B NNI swapping is a basic step in topology building and searching A D C B e

Tree Length Formula With T as in prior slide, Using lemma and branch length formula for l(e),

Tree Length after NNI Given T  T ’ the tree swap in prior slide, l the edge length function: where l and l ’ are constants depending on the topologies. (1)

OLS: FASTNNI 1. Pre-compute average distances between non- intersecting sub-trees. ( O ( n 2 ) computations) 2. Loop over all internal edges, select the best swap using Equation (1). ( O ( n )) 3. If no swap improves length of the tree, stop and return the tree, else perform the best swap and update the matrix of average distances and repeat Step 2. ( O ( n ) per swap; there is only one new split.) Thus, if we require p swaps, the total complexity of FASTNNI is O(n 2 + pn).

Balanced Minimum Evolution  Gascuel (2000) observed that the OLS/ME method was weaker than NJ in approximating the correct topology.  Pauplin (2000) to simplify tree length computation proposed to use a “balanced” version of Minimum Evolution, weighting each sub-tree equally when calculating averages: if A and B are sub-trees of T, with

BNNI 1. Calculate balanced averages of all pairs of sub-trees. ( O( n 2 ) ) 2. Calculate improvement for each swap using (2) 3. If no tree swap improves length of the tree, stop and return tree, else update matrix of average distances and repeat Step 2. ( O ( n diam ( T )) per swap) The average complexity, when performing p swaps, is O( n 2 + pn diam(T)).

Updating Subtree Averages e C DB y x X A T Y Q: How many recalculations? If we perform the B-C tree swap, then we must recalculate Typical values for diam( T ): Yule-Harding distribution: Uniform distribution: (Hint: you can count (x,y) pairs). A: O( n diam( T))

Building trees from scratch We have NNI algorithms for OLS and balanced branch lengths. But what if we have no initial topology for NNIs?

OLS: Greedy Minimum Evolution 1. Start with three-taxon tree T 3 2. For k=4 to n, a)Calculate D k|A for each subtree A in T k-1 b)Express cost of inserting k along edge e as f(e). (Use Equation (3) on the next slide.) c)Choose e minimizing f. Insert k along e to form T k. d)Update matrix of average distances between every pair of 2-distant subtrees. GME runs in O(n 2 ) running time

Greedy Minimum Evolution C AB k T C AB k T’T’ Then We use a variant of Equation (1), where D = {k}. Let L = l(T).

Balanced Minimum Evolution Same as GME,except: 2. (modifications) a) Calculate balanced average distances instead of ordinary average distances b) Use l = ½ to find weights for insertion points d) Must keep average distances for all pairs of sub- trees. BME runs in O(n 2 diam(T)) running time.

Simulations  Created 24- and 96-taxon trees, 2000 per each size, Yule-Harding process (  molecular clock).  Edge lengths multiplied by (1.0 + m X), where X is exponentially distributed.  Generated trees with three rates of evolution  SeqGen used to generate sequences for each tree and rate (12,000 in all)  DNADIST used to calculate distance matrices

Results: topological distances BNNI improved all input trees

Results: topological distances This improvement is large with fast rates and high numbers of taxa

Results: topological distances NNI trees are close to the best possible for BME

Results: topological distances The quality of the NNI tree is (mostly) independent of starting point

Results: topological distances FASTNNI trees comparable to NJ as n grows to 96

Computational Times 24 Taxa96 Taxa1000 Taxa4000 Taxa GME + BNNI :02.1 HGT/FP :33.1 NJ/BIONJ :55.9 WEIGHBOR FITCH Computations done on Sun Enterprise E4500/E5500 running Solaris 8 on Mhz processors with 7 Gb memory. in (MM:SS)

Average number of NNIs We see that the average number of NNIs is considerably lower than the number of taxa. 24 Taxa96 Taxa1000 Taxa4000 Taxa GME + FASTNNI GME + BNNI BME + BNNI

BME = WLS Why does the balanced approach work so well?  Pauplin’s formula for the length of a tree is  BME is a weighted least squares approach with Where p T (i,j) is the length of the (i,j) path in T. Distantly related taxa see their importance decrease exponentially.

Bonus features  BME is a consistent method. As observed distances converge to true distances, the true topology becomes the minimum evolution tree.  The BNNI tree has no negative branch lengths. A negative value to the branch length function implies a NNI leading to a smaller tree.

Consistency of Balanced ME  Theorem : Suppose S is a weighted tree, and T is a tree topology incompatible with S. Let T be the tree of topology T with weights determined by the balanced scheme. Then l(T) > l(S).  Lemma: it suffices to prove the case when S is a split metric.

Balanced ME consistency  Basic idea: let l be the tree length function on the space of topologies. We find a sequence of topologies, T=T 0, T 1,... T k =S such that Each T i+1 can be reached from T i via one of two simple topological transformations l(T i ) > l(T i+1 ) for all i. Proof structure modeled after OLS/ME proof (Rzhetsky and Nei, 1993).

Type I transformation B C C AA DD B Color the leaves black or white according to the split metric S. A Type I transformation uses a NNI to form a larger monochromatic cluster This transformation reduces the size of the tree under l

Type II transformation A1A1 B1B1 A1A1 B1B1 B2B2 B2B2 A2A2 A2A2 C C A Type II transformation uses two NNIs to form two monochromatic subtrees This transformation also reduces the value of the size of the tree under l

Positive Branch Lengths after BNNI Recall that the length of an edge is described by C e A D B We do not perform the switch because i.e. Similarly, Thus

Conclusions  BME + BNNI runs in O((n 2 + pn) diam(T)), outputs trees comparable to (better than) FITCH, Weighbor, BioNJ, or NJ.  FastME is faster than NJ or its variants.  BNNI consistently improved output trees in all settings, even when WLS/Fitch trees were input.  BNNI outputs tree without negative branch lengths.  FASTME software available at or