Fast Algorithms for Minimum Evolution

1
Fast Algorithms for Minimum Evolution

• Richard Desper, NCBI
• Olivier Gascuel, LIRMM

2
Overview

1. Statement of phylogeny reconstruction problem and
   various approaches to solving it.
2. Tree length formula as a function of average
   distances.
3. Greedy algorithms for tree building and tree
   swapping.
4. Simulation results.
5. A few extras regarding consistency and branch
   lengths.

3
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.

4
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.

5
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, DT is
  the induced tree metric)

If sij 1 for all i and j, this is called the
ordinary least-squares method.
6
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 )

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Recursive Definition of DAB

• If A a, B b, DAB Dab,

All average distances for all pairs of
non-intersecting subtrees of a given topology
can be calculated in O(n2) time.
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External OLS Edge Length Function

• If e is the edge connecting the leaf i to the
• subtrees A and B,

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Internal OLS Edge Length Function

• The length of the edge e is (Vach, 1988)

where
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Tree length formula

• Lemma with T as to the right,
• let denote the root of subtree X,
• and the edge to X for
• Then,

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Tree Length Formula

• With T as in prior slide,

Using lemma and branch length formula for l(e),
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General approach

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

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Tree Swapping by NNI
NNI swapping is a basic step in topology building
and searching
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Tree Length Formula

• With T as in prior slide,

Using lemma and branch length formula for l(e),
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Tree Length after NNI

• Given T g T the tree swap in prior slide, l
  the edge length function

(1)
where l and l are constants depending on the
topologies.
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OLS FASTNNI

• Pre-compute average distances between
  non-intersecting sub-trees. (O(n2) computations)
• Loop over all internal edges, select the best
  swap using Equation (1). (O(n))
• 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(n2 pn).

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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

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BNNI

• Calculate balanced averages of all pairs of
  sub-trees. (O(n2))
• Calculate improvement for each swap using
• (2)
• 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(n2 pn diam(T)).

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Updating Subtree Averages
T
x
X
A
C
e
Y
D
B
Q How many recalculations?
(Hint you can count (x,y) pairs).
A O(n diam(T))
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Building trees from scratch

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

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OLS Greedy Minimum Evolution

• Start with three-taxon tree T3
• For k4 to n,
• Calculate DkA for each subtree A in Tk-1
• Express cost of inserting k along edge e as f(e).
  
• (Use Equation (3) on the next slide.)
• Choose e minimizing f. Insert k along e to form
  Tk.
• Update matrix of average distances between every
  pair of 2-distant subtrees.
• GME runs in O(n2) running time

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Greedy Minimum Evolution
We use a variant of Equation (1), where D k.
Let L l(T).
Then
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Balanced Minimum Evolution

• Same as GME,except
• (modifications)
• Calculate balanced average distances instead of
  ordinary average distances
• Use l ½ to find weights for insertion points
• Must keep average distances for all pairs of
  sub-trees.
• BME runs in O(n2 diam(T)) running time.

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Simulations

• Created 24- and 96-taxon trees, 2000 per each
  size, Yule-Harding process (g molecular clock).
• Edge lengths multiplied by (1.0 mX), 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

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Results topological distances
BNNI improved all input trees
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Results topological distances
This improvement is large with fast rates and
high numbers of taxa
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Results topological distances
NNI trees are close to the best possible for BME
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Results topological distances
The quality of the NNI tree is (mostly) independen
t of starting point
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Results topological distances
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Computational Times
in (MMSS)
24 Taxa 96 Taxa 1000 Taxa 4000 Taxa
GME BNNI 0.0263 0.0842 11.3390 0602.1
HGT/FP 0.0252 0.1349 13.8080 0333.1
NJ/BIONJ 0.0630 0.1628 21.2500 2055.9
WEIGHBOR 0.4244 26.8818    
FITCH 4.3745      
Computations done on Sun Enterprise E4500/E5500
running Solaris 8 on 10 400-Mhz processors with 7
Gb memory.
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Average number of NNIs
24 Taxa 96 Taxa 1000 Taxa 4000 Taxa
GME FASTNNI 1.244 8.446 44.9 336.50
GME BNNI 1.446 11.177 59.1 343.75
BME BNNI 1.070 6.933 29.1 116.25
We see that the average number of NNIs is
considerably lower than the number of taxa.
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BME WLS

• Why does the balanced approach work so well?
• Pauplins formula for the length of a tree is
• BME is a weighted least squares approach with

Where pT(i,j) is the length of the (i,j) path in
T.
Distantly related taxa see their importance
decrease exponentially.
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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.

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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) gt l(S).
• Lemma it suffices to prove the case when S is a
  split metric.

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Balanced ME consistency

• Basic idea let l be the tree length function on
  the space of topologies. We find a sequence of
  topologies, TT0, T1, ... TkS such that
• Each Ti1 can be reached from Ti via one of two
  simple topological transformations
• l(Ti) gt l(Ti1) for all i.
• Proof structure modeled after OLS/ME proof
  (Rzhetsky and Nei, 1993).

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Type I transformation
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
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Type II transformation
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
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Positive Branch Lengths after BNNI

• Recall that the length of an
  edge is described by

We do not perform the switch because
i.e.
Thus
Similarly,
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Conclusions

• BME BNNI runs in O((n2 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 http//www.ncbi.nlm.n
  ih.gov/CBBResearch/Desper/FastME.html or
  http//www.lirmm.fr/w3ifa/MAAS/.