Phylogenetic trees III Maximum Parsimony Gerhard Jger Words, - - PowerPoint PPT Presentation

Phylogenetic trees III Maximum Parsimony

Gerhard Jäger Words, Bones, Genes, Tools February 28, 2018

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Background

Background

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Background

Character-based tree estimation

distance-based tree estimation has several drawbacks:

very strong theoretical assumptions - e.g., all characters evolve at the same rate Neighbor Joining and UPGMA produce good but sub-optimal trees no solid statistical justifjcation for those algorithms distances are black boxes — we get a tree, but we learn nothing about the history of individual characters

character-based tree estimation

estimates complete scenario (or distribution over scenarios) for each character fjnds the tree that best explains the observed variation in the data (at least in theory, that is...)

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Parsimony

Parsimony

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Parsimony

Parsimony of a tree

background reading: Ewens and Grant (2005), 15.6 suppose a character matrix and a tree are given parsimony score: minimal number of mutations that has to be assumed to explain the character values at the tips, given the tree

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Parsimony

Parsimony of a tree

Kopf "head" kop "head" head "head" tête "head" testa "head" cap "head"

Gerhard Jäger Maximum Parsimony WBGT 6 / 30

Parsimony

Parsimony of a tree

Kopf "head" kop "head" head "head" tête "head" testa "head" cap "head"

Gerhard Jäger Maximum Parsimony WBGT 6 / 30

Parsimony

Parsimony of a tree

Kopf "head" kop "head" head "head" tête "head" testa "head" cap "head" "head"

? ? ? ? ?

Gerhard Jäger Maximum Parsimony WBGT 6 / 30

Parsimony

Parsimony of a tree

Kopf "head" kop "head" head "head" tête "head" testa "head" cap "head" *kop "head" testa "head"

Gerhard Jäger Maximum Parsimony WBGT 6 / 30

Parsimony

Parsimony of a tree

Kopf "head" kop "head" head "head" tête "head" testa "head" cap "head" *kop "head" *haubud- "head" testa "head" caput "head" *kaput- "head"

Gerhard Jäger Maximum Parsimony WBGT 6 / 30

Parsimony

Parsimony reconstruction

A C C B A B B A B B C

Parsimony = 2

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Parsimony

Parsimony reconstruction

A C C A B B A B C

Parsimony = 3

A A

Gerhard Jäger Maximum Parsimony WBGT 7 / 30

Parsimony

Parsimony reconstruction

A C C A B B A C

Parsimony = 3

A C C

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Parsimony

Weighted parsimony reconstruction

A C C B A B B A B B C Weighted Parsimony = 3

Weight matrix A B C A 1 2 B 1 2 C 2 2

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Parsimony

Weighted parsimony reconstruction

A C C A B B A B C A A Weighted Parsimony = 4

Weight matrix A B C A 1 2 B 1 2 C 2 2

Gerhard Jäger Maximum Parsimony WBGT 8 / 30

Parsimony

Weighted parsimony reconstruction

A C C A B B A C Weighted Parsimony = 5 A C C

Weight matrix A B C A 1 2 B 1 2 C 2 2

Gerhard Jäger Maximum Parsimony WBGT 8 / 30

Parsimony

Dynamic Programming (Sankofg Algorithm)

wp(mother, s) =

• d∈daughters

min

s′∈states(w(s, s′) + wp(d, s′))

A C C A B B

0 ∞ ∞ 0 ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ Gerhard Jäger Maximum Parsimony WBGT 9 / 30

Parsimony

Dynamic Programming (Sankofg Algorithm)

wp(mother, s) =

• d∈daughters

min

s′∈states(w(s, s′) + wp(d, s′))

A C C A B B

0 ∞ ∞ 0 ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

0 2 4 4 4 0

Gerhard Jäger Maximum Parsimony WBGT 9 / 30

Parsimony

Dynamic Programming (Sankofg Algorithm)

wp(mother, s) =

• d∈daughters

min

s′∈states(w(s, s′) + wp(d, s′))

A C C A B B

0 ∞ ∞ 0 ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

1 3 2 4 1 4 4 4 0 2 2

Gerhard Jäger Maximum Parsimony WBGT 9 / 30

Parsimony

Dynamic Programming (Sankofg Algorithm)

wp(mother, s) =

• d∈daughters

min

s′∈states(w(s, s′) + wp(d, s′))

A C C A B B

0 ∞ ∞ 0 ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

1 3 4 2 4 1 4 4 4 0 2 2 3 5

Gerhard Jäger Maximum Parsimony WBGT 9 / 30

Parsimony

Dynamic Programming (Sankofg Algorithm)

wp(mother, s) =

• d∈daughters

min

s′∈states(w(s, s′) + wp(d, s′))

A C C A B B

0 ∞ ∞ 0 ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

1 3 4 2 4 1 4 4 4 0 2 2 3 5

Gerhard Jäger Maximum Parsimony WBGT 10 / 30

Parsimony

Dynamic Programming (Sankofg Algorithm)

wp(mother, s) =

• d∈daughters

min

s′∈states(w(s, s′) + wp(d, s′))

A C C A B B

0 ∞ ∞ 0 ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

1 3 4 2 4 1 4 4 4 0 2 2 3 5

Gerhard Jäger Maximum Parsimony WBGT 10 / 30

Parsimony

Dynamic Programming (Sankofg Algorithm)

wp(mother, s) =

• d∈daughters

min

s′∈states(w(s, s′) + wp(d, s′))

A C C A B B

0 ∞ ∞ 0 ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

1 3 4 2 4 1 4 4 4 0 2 2 3 5

Gerhard Jäger Maximum Parsimony WBGT 10 / 30

Parsimony

Dynamic Programming (Sankofg Algorithm)

wp(mother, s) =

• d∈daughters

min

s′∈states(w(s, s′) + wp(d, s′))

A C C A B B

0 ∞ ∞ 0 ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

1 3 4 2 4 1 4 4 4 0 2 2 3 5

Gerhard Jäger Maximum Parsimony WBGT 10 / 30

Parsimony

Dynamic Programming (Sankofg Algorithm)

wp(mother, s) =

• d∈daughters

min

s′∈states(w(s, s′) + wp(d, s′))

A C C A B B

0 ∞ ∞ 0 ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

1 3 4 2 4 1 4 4 4 0 2 2 3 5

Gerhard Jäger Maximum Parsimony WBGT 10 / 30

Parsimony

Searching for the best tree

total parsimony score of tree: sum over all characters note: if weight matrix is symmetric, location of the root doesn’t matter Sankofg algorithm effjciently computes parsimony score of a given tree goal: tree which minimizes parsimony score no effjcient way to fjnd the optimal tree → heuristic tree search

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Searching the tree space

Searching the tree space

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Searching the tree space

How many rooted tree topologies are there?

2 1

n=2

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Searching the tree space

How many rooted tree topologies are there?

2 3 1 3 2 1 2 3 1 2 1

n=2 n=3

2 3 1

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Searching the tree space

How many rooted tree topologies are there?

2 3 1 2 4 3 1 2 4 3 1 2 4 3 1 2 3 4 1 3 2 1 2 3 1 2 1

n=2 n=3 n=4

2 3 1 2 4 3 1

Gerhard Jäger Maximum Parsimony WBGT 13 / 30

Searching the tree space

How many rooted tree topologies are there?

f(2) = 1 f(n + 1) = (2n − 3)f(n) f(n) = (2n − 3)! 2n−2(n − 2)!

2 1 3 3 4 15 5 105 6 945 7 10395 8 135135 9 2027025 10 34459425 11 654729075 12 13749310575 13 316234143225 14 7.9e + 12 15 2.1e + 14 16 6.1e + 15 17 1.9e + 17 18 6.3e + 18 19 2.2e + 20 20 8.2e + 21 21 3.1e + 23 22 1.3e + 25 23 5.6e + 26 24 2.5e + 28 25 1.1e + 30 26 5.8e + 31 27 2.9e + 33 28 1.5e + 35 29 8.6e + 36 30 4.9e + 38 31 2.9e + 40 32 1.7e + 42 33 1.1e + 44 34 7.2e + 45 35 4.8e + 47 36 3.3e + 49 37 2.3e + 51 38 1.7e + 53 39 1.3e + 55 40 1.0e + 57

Gerhard Jäger Maximum Parsimony WBGT 14 / 30

Searching the tree space

How many unrooted tree topologies are there?

2 3 1

n=3 Gerhard Jäger Maximum Parsimony WBGT 15 / 30

Searching the tree space

How many unrooted tree topologies are there?

2 3 4 1 3 2 4 1 4 2 3 1 2 3 1

n=3 n=4 Gerhard Jäger Maximum Parsimony WBGT 15 / 30

Searching the tree space

How many unrooted tree topologies are there?

2 5 4 3 1 2 3 4 5 1 2 4 5 3 1 2 5 3 4 1 5 2 3 4 1 3 5 2 4 1 2 4 3 5 1 3 2 5 4 1 3 4 2 5 1 5 3 4 2 1 4 5 2 3 1 4 2 3 5 1 4 3 5 2 1 4 5 2 3 1 5 4 2 3 1 2 3 4 1 3 2 4 1 4 2 3 1 2 3 1

n=3 n=4 n=5 Gerhard Jäger Maximum Parsimony WBGT 15 / 30

Searching the tree space

How many unrooted tree topologies are there?

f(3) = 1 f(n + 1) = (2n − 3)f(n) f(n) = (2n − 5)! 2n−3(n − 3)!

3 1 4 3 5 15 6 105 7 945 8 10395 9 135135 10 2027025 11 34459425 12 654729075 13 13749310575 14 316234143225 15 7.90e + 12 16 2.13e + 14 17 6.19e + 15 18 1.91e + 17 19 6.33e + 18 20 2.21e + 20 21 8.20e + 21 22 3.19e + 23 23 1.31e + 25 24 5.63e + 26 25 2.53e + 28 26 1.19e + 30 27 5.84e + 31 28 2.98e + 33 29 1.57e + 35 30 8.68e + 36 31 4.95e + 38 32 2.92e + 40 33 1.78e + 42 34 1.12e + 44 35 7.29e + 45 36 4.88e + 47 37 3.37e + 49 38 2.39e + 51 39 1.74e + 53 40 1.31e + 55

Gerhard Jäger Maximum Parsimony WBGT 16 / 30

Searching the tree space

Heuristic tree search

tree space is too large to do an exhaustive search if n (number of taxa) is larger than 12 or so heuristic search:

start with some tree topology (e.g., Neighbor-Joining tree) apply a bunch of local modifjcations to the current tree if one of the modifjed tree has lower or equal parsimony, move to that tree stop if no further improvement is possible

⇒ standard approach for optimization problems in computer science

Gerhard Jäger Maximum Parsimony WBGT 17 / 30

Searching the tree space

Tree modifjcations

three tree modifjcations commonly in use:

1

Nearest Neighbor Interchange (NNI)

2

Tree Bisection and Reconnection (TBR)

3

Subtree Pruning and Regrafting (SPR)

local modifjcations are better than arbitrary moves in tree space because partial parsimony computations can be re-used in modifjed tree

Gerhard Jäger Maximum Parsimony WBGT 18 / 30

Searching the tree space

Nearest Neighbor Interchange

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Searching the tree space

Tree Bisection and Reconection

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 10 9 1 2 3 4 5 6 7 8 10 9

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Searching the tree space

Subtree Pruning and Regrafting

1 2 3 4 5 6 7 8 10 9 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 10 9

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Searching the tree space

Heuristic tree search

NNI is very local → only O(n) possible moves SPR and TBR are more aggressive → O(n2)/O(n3) possible moves NNI search is comparatively fast, but prone to get stuck in local

• ptima

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Searching the tree space

Running example: SPR search with cognate data

parsimony=1984

Spanish Portuguese Hindi N e p a l i B e n g a l i Greek B r e t

• n

Welsh Irish Dutch G e r m a n E n g l i s h Swedish Danish Icelandic Polish Czech Russian U k r a i n i a n B u l g a r i a n Lithuanian French Italian Romanian C a t a l a n

starting with Neighbor Joining tree . . .

Gerhard Jäger Maximum Parsimony WBGT 23 / 30

Searching the tree space

Running example: SPR search with cognate data

parsimony=1979

S p a n i s h Portuguese Hindi N e p a l i B e n g a l i Greek Breton W e l s h Irish Dutch G e r m a n English Swedish D a n i s h Icelandic Polish Czech R u s s i a n Ukrainian Bulgarian Lithuanian French Italian R

• m

a n i a n Catalan

Gerhard Jäger Maximum Parsimony WBGT 23 / 30

Searching the tree space

Running example: SPR search with cognate data

parsimony=1975

Spanish P

• r

t u g u e s e H i n d i Nepali B e n g a l i Greek B r e t

• n

Welsh Irish Dutch G e r m a n English Swedish D a n i s h Icelandic Polish Czech Russian U k r a i n i a n Bulgarian Lithuanian French Italian Romanian C a t a l a n

Gerhard Jäger Maximum Parsimony WBGT 23 / 30

Searching the tree space

Running example: SPR search with cognate data

parsimony=1973

Spanish Portuguese H i n d i Nepali B e n g a l i Greek Breton W e l s h Irish D u t c h German English S w e d i s h Danish Icelandic Polish Czech R u s s i a n Ukrainian Bulgarian Lithuanian F r e n c h Italian Romanian C a t a l a n

Gerhard Jäger Maximum Parsimony WBGT 23 / 30

Searching the tree space

Running example: SPR search with cognate data

parsimony=1969

Spanish Portuguese H i n d i Nepali B e n g a l i Greek Breton W e l s h Irish D u t c h German English S w e d i s h Danish Icelandic Polish Czech R u s s i a n Ukrainian Bulgarian Lithuanian F r e n c h Italian Romanian C a t a l a n

. . . Maximum Parsimony tree

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Searching the tree space

Running example: SPR search with cognate data

there are actually 16 difgerent trees with minimal parsimony score

Greek Irish Welsh Breton French Italian Catalan Portuguese Spanish Romanian Icelandic Swedish Danish German Dutch English Hindi Nepali Bengali Ukrainian Russian Polish Czech Bulgarian Lithuanian

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Searching the tree space

MP tree for WALS characters

Bengali Breton Irish Welsh Bulgarian Greek Czech Lithuanian Polish Russian Ukrainian Catalan Romanian Italian Spanish Portuguese French Danish Swedish Icelandic Dutch German English Hindi Nepali

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Searching the tree space

MP tree for sound-concept characters

Greek B u l g a r i a n Russian Ukrainian Polish Czech Lithuanian Icelandic S w e d i s h Danish D u t c h German E n g l i s h C a t a l a n Portuguese Spanish I t a l i a n Romanian French Breton Welsh Irish Hindi Bengali Nepali

Gerhard Jäger Maximum Parsimony WBGT 26 / 30

Searching the tree space

Dollo parsimony

previous trees were estimated with a symmetric weight matrix if weights are asymmetric, location of the root matters extreme case: Dollo Parsimony w(0 → 1) = ∞

Spanish Portuguese Hindi Nepali Bengali Greek Breton Welsh Irish Dutch German English Swedish Danish Icelandic Polish Czech Russian Ukrainian Bulgarian Lithuanian French Italian Romanian Catalan

Gerhard Jäger Maximum Parsimony WBGT 27 / 30

Searching the tree space

Maximum Parsimony: Discussion

Once we have found the best tree (or, in any event, which is very close to the best tree), we can reconstruct ancestral states via the Sankofg algorithm this allows to compute statistics about stability of characters, frequency and location of parallel changes etc.

⇒ much more informative than distance-based inference

Gerhard Jäger Maximum Parsimony WBGT 28 / 30

Searching the tree space

Maximum Parsimony: Discussion

disadvantages of MP:

simulation studies: capacity to recover the true tree is decent but not

• verwhelming

possibility of multiple mutations on a single branch is not taken into consideration all characters are treated equal; no discrimination between stable and volatile characters ties are common, especially if you have few data values for weight matrix are ad hoc no real theoretical justifjcation

Why should the true tree minimize the total number of mutations? Rests on a valid intuition: Mutations are unlikely, so assuming fewer mutations increases the likelihood of the data. Likelihood is not formally derived from a probabilistic modell though.

Next step: Maximum Likelihood tree estimation

Gerhard Jäger Maximum Parsimony WBGT 29 / 30

Searching the tree space

Hands on

Install the software Paup*. Go to the directory where you have the put the nexus fjles and type > paup4 ielex.bin.nex At Paup’s command prompt, type paup> hsearch. Display tree with paup> describetree /plot=phylo Save result with paup> savetree format=newick file = ielex.mp.tre \ brlen=yes Leave Paup* with paup> q Install Dendroscope or FigTree and load ielex.mp.tre.

Gerhard Jäger Maximum Parsimony WBGT 30 / 30

Searching the tree space

Ewens, W. and G. Grant (2005). Statistical Methods in Bioinformatics: An Introduction. Springer, New York.

Gerhard Jäger Maximum Parsimony WBGT 30 / 30