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

Fast Algorithms for Minimum EvolutionFast Algorithms for Minimum Evolution

Richard Desper, NCBIOlivier Gascuel, LIRMM

OverviewOverview

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 ReconstructionPhylogeny 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.

MethodsMethods

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

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 ( is the input metric, T is the induced tree metric):

2

2

,

Tij ij

iji j

If ij = 1 for all i and j, this is called the ordinary least-squares method.

Minimum EvolutionMinimum Evolution Developed by Rzhetsky and Nei (1992) as a

modification of the OLS method For each topology ,

Define function l assigning OLS lengths to edges of

Define size of tree

Choose minimizing l()

( )

( ) ( )e E

l l e

T

T

Recursive Definition of A|BRecursive Definition of A|B

If A = {a}, B = {b}, A|B = ab,

1 2

1 2| | |

1 2 1 2A B A B A B

B B

B B B B

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

For 1 2 ,B B B 1B

2B

A

External OLS Edge Length Function

External OLS Edge Length Function

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

| | |

1( )

2 Ai B i A Bl e

e

B

A

i

Internal OLS Edge Length Function

Internal OLS Edge Length Function

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

eA C

DB

| | | | | |

1( ) ( ) (1 )( ) ( )

2 A C B D A D B C A B B Cl e

where| || | | || |

.(| | | |)(| | | |)

A D B C

A B C D

Tree length formulaTree 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

Xr

{ , , , }.X A B C D

| | |( ) ( )A BA B A r A B B rl e l e

Xe

Tree Length FormulaTree Length Formula

With T as in prior slide,

| |

| | |, , ,

| |

1( ) ( ) (1 )

2X

AC B D

X r A D B CX A B C D

A B C D

l T l X

, , ,

( ) ( ) ( ) ( )XX A B C D

l T l e l X l e

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

General approachGeneral approach

To search the space of topologies, we’ll keep in memory two data structures:

Sizes of each subtree of given topologyMatrix 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 NNITree 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 FormulaTree Length Formula

With T as in prior slide,

| |

| | |, , ,

| |

1( ) ( ) (1 )

2X

AC B D

X r A D B CX A B C D

A B C D

l T l X

, , ,

( ) ( ) ( ) ( )XX A B C D

l T l e l X l e

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

Tree Length after NNITree Length after NNI

Given ’ the tree swap in prior slide, l the edge length function:

| |

' '| |

'| |

( 1)( )1

( ) ( ) ( 1)( )2

( )( )

AC B D

A B C D

A D B C

l l

T T

where and ’ are constants depending on the topologies.

(1)

OLS: FASTNNIOLS: FASTNNI1. Pre-compute average distances between non-

intersecting sub-trees. (O(n2) 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 ofFASTNNI is O(n2 + 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 , with

1 2| | |

1 1

2 2A B A B A B T T T

1 2 ,B B B

BNNIBNNI

1. Calculate balanced averages of all pairs of sub-trees. (O(n2))

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, isO(n2 + pn diam(T)).

| | | |

1( ) ( ')

2 A B C D A C B Dl T l T T T T T

Updating Subtree Averages

Updating Subtree Averages

e

C

DB

y

xX

A

T

Y

Q: How many recalculations?

If we perform the B-C tree swap, then we must recalculate |

TX Y

...X AHere, B C D Y ...and

Typical values for diam(T):

Yule-Harding distribution:

Uniform distribution:

(log )O n

O n

(Hint: you can count (x,y) pairs).

A: O(n diam(T))

Building trees from scratchBuilding 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

OLS: Greedy Minimum Evolution

1. Start with three-taxon tree T3

2. For k=4 to n, a) Calculate k|A for each subtree A in Tk-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 Tk.d) Update matrix of average distances between every pair

of 2-distant subtrees.GME runs in O(n2) running time

Greedy Minimum Evolution

Greedy Minimum Evolution

C

A B

k

T C

A B

k

T’

' | |

' '| |

| |

( )1

(3) ( ) ( 1)( )2

(1 )( )

k A B C

A B k C

A C k B

l T L

Then

We use a variant of Equation (1), where D = {k}. Let L = l(T).



Same as GME,except:2. (modifications)

d) Calculate balanced average distances instead of ordinary average distances

e) Use = ½ to find weights for insertion pointsf) Must keep average distances for all pairs of sub-

trees.

BME runs in O(n2 diam(T)) running time.

SimulationsSimulations Created 24- and 96-taxon trees, 2000 per each

size, Yule-Harding process ( molecular clock). Edge lengths multiplied by (1.0 + 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


0.100

0.105

0.110

0.115

0.120

TRUE

FITCH

NJ/W

LS

WEIG

HBOR

BIONJ

NJBM

EGM

E

HGT/

FP

with FastNNI

without NNIs

with BNNI

24-Taxa Trees - Slow Rate of Evolution

BNNI improvedall input trees



0.070

0.080

0.090

0.100

0.110

0.120

TRUE

NJ/FI

TCH

WEIG

HBOR

BIONJ

NJBM

EGM

E

HGT/

FP

with FastNNI

without NNIs

with BNNI

96-Taxa Trees - Fast Rate of Evolution

This improvementis large with fast rates and high numbers of taxa



0.100

0.105

0.110

0.115

0.120

TRUE

FITCH

NJ/W

LS

WEIG

HBOR

BIONJ

NJBM

EGM

E

HGT/

FP

with FastNNI

without NNIs

with BNNI


NNI trees are close to the best possible for BME



0.100

0.105

0.110

0.115

0.120

TRUE

FITCH

NJ/W

LS

WEIG

HBOR

BIONJ

NJBM

EGM

E

HGT/

FP

with FastNNI

without NNIs

with BNNI


The quality of theNNI tree is (mostly)independentof starting point



0.17

0.175

0.18

0.185

0.19

0.195

0.2

with FastNNI

without NNIs

with BNNI


FASTNNI trees comparable to NJ as n grows to 96

Computational TimesComputational Times

24 Taxa 96 Taxa 1000 Taxa 4000 Taxa

GME + BNNI 0.0263 0.0842 11.3390 06:02.1

HGT/FP 0.0252 0.1349 13.8080 03:33.1

NJ/BIONJ 0.0630 0.1628 21.2500 20:55.9

WEIGHBOR 0.4244 26.8818

FITCH 4.3745

Computations done on Sun Enterprise E4500/E5500 running Solaris 8on 10 400-Mhz processors with 7 Gb memory.

in (MM:SS)

Average number of NNIsAverage number of NNIs

We see that the average number of NNIs is considerably lower than the number of taxa.

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

BME = WLSBME = 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

1 ( , )( ) 2 ,Tp i j

iji j

l T

Where pT(i,j) is the length of the (i,j) path in T.

( , ).Tij cp i j Distantly related taxa see their importance

decrease exponentially.

Bonus featuresBonus 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

Consistency of Balanced ME

Theorem: Suppose S is a weighted tree, and is a tree topology incompatible with S. Let T be the tree of topology 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 consistencyBalanced ME consistency

Basic idea: let l be the tree length function on the space of topologies. We find a sequence of topologies, T=T0, T1, ... Tk=S such that

Each Ti+1 can be reached from Ti via one of two simple topological transformationsl(Ti) > l(Ti+1) for all i.

Proof structure modeled after OLS/ME proof (Rzhetsky and Nei, 1993).

Type I transformationType I transformation

BC C

AAD D

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 transformationType II transformation

A1

B1

A1

B1

B2 B2

A2

A2

C C

A Type II transformation uses two NNIs to form two monochromaticsubtrees

This transformation also reduces the value of the size of the treeunder l

Positive Branch Lengths after BNNI

Positive Branch Lengths after BNNI

Recall that the length of an edge is described by

| | | | | |

1 1( ) ( ) ( )

2 2 AC B D A D B C A B C Dl e

| | | |

1( ) ( ') 0,

2 A B C D A C B Dl T l T T T T T

C

e

A

D

B

We do not perform the switch becauseB C

i.e. | | | | .A C B D A B C D

Similarly, | | | | .A D B C A B C D Thus ( ) 0l e

ConclusionsConclusions 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.nih.gov/CBBResearch/Desper/FastME.html or http://www.lirmm.fr/~w3ifa/MAAS/.

Documents

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