Tree searching methods
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Tree Searching Methods. Exhaustive search (exact) Branch-and-bound search (exact) Heuristic search methods (approximate) Stepwise addition Branch swapping Star decomposition. Exhaustive Search. 12. 12. 13. 13. 13. 12. 13. 13. 12. 13. 11. 13. 13. 13. 13. Searching for trees.

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Tree Searching Methods

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Tree searching methods

Tree Searching Methods

  • Exhaustive search (exact)

  • Branch-and-bound search (exact)

  • Heuristic search methods (approximate)

    • Stepwise addition

    • Branch swapping

    • Star decomposition


Exhaustive search

Exhaustive Search

12

12

13

13

13

12

13

13

12

13

11

13

13

13

13


Searching for trees

Searching for trees

1.Generate all 3 trees for first 4 taxa:

  • Generation of all possible trees


Searching for trees1

Searching for trees

2. Generate all 15 trees for first 5 taxa:

(likewise for each of the other two 4-taxon trees)


Searching for trees2

Searching for trees

3. Full search tree:


Searching for trees3

Searching for trees

The search tree is the same as for exhaustive search, with tree lengths for a hypothetical data set shown in boldface type. If a tree lying at a node of this search tree has a length that exceeds the current lower bound on the optimal tree length, this path of the search tree is terminated (indicated by a cross-bar), and the algorithm backtracks and takes the next available path. When a tip of the search tree is reached (i.e., when we arrive at a tree containing the full set of taxa), the tree is either optimal (and hence retained) or suboptimal (and rejected). When all paths leading from the initial 3-taxon tree have been explored, the algorithm terminates, and all most-parsimonious trees will have been identified. Asterisks indicate points at which the current lower bound is reduced. Circled numbers represent the order in which phylogenetic trees are visited in the search tree.

Branch and bound algorithm:


Stepwise addition in a nutshell

1

2

2

1

1

3

4

3

3

4

4

2

Stepwise Addition (in a nutshell)

2

1

3


Searching for trees4

Searching for trees

Stepwise addition

A greedy stepwise-addition search applied to the example used for branch-and-bound. The best 4-taxon tree is determined by evaluating the lengths of the three trees obtained by joining taxon D to tree 1 containing only the first three taxa. Taxa E and F are then connected to the five and seven possible locations, respectively, on trees 4 and 9, with only the shortest trees found during each step being used for the next step. In this example, the 233-step tree obtained is not a global optimum. Circled numbers indicate the order in which phylogenetic trees are evaluated in the stepwise-addition search.


Stepwise addition variants

Stepwise Addition Variants

  • As Is

    • add in order found in matrix

  • Closest

    • add unplaced taxa that requires smallest increase

  • Furthest

    • add unplaced taxa that requires largest increase

  • Simple

    • Farris’s (1970) “simple algorithm” uses a set of pairwise reference distances

  • Random

    • random permutation of taxa is used to select the order


Branch swapping nearest neighbor interchange nni

B

C

E

A

D

E

D

A

B

C

Branch swappingNearest Neighbor Interchange (NNI)

B

C

D

A

E


Branch swapping subtree pruning and regrafting spr

B

A

E

B

D

A

C

F

C

G

D

D

C

E

E

A

F

F

B

G

G

Branch swappingSubtree Pruning and Regrafting (SPR)

a


Branch swapping tree bisection and reconnection tbr

E

D

B

A

F

G

B

A

A

C

C

E

D

B

C

D

D

F

G

E

E

E

D

C

F

A

F

F

G

B

G

G

B

A

C

Branch swappingTree Bisection and Reconnection (TBR)


Reconnection limits in tbr

Reconnection limits in TBR

Reconnection distances:


Tree searching methods

Reconnection limits in TBR

Reconnection distances:

In PAUP*, use “ReconLim” to set maximum reconnection distance


Tree searching methods

Star-decomposition search


Overview of maximum likelihood as used in phylogenetics

Overview of maximum likelihood as used in phylogenetics

  • Overall goal: Find a tree topology (and associated parameter estimates) that maximizes the probability of obtaining the observed data, given a model of evolution

    Likelihood(hypothesis) µProb(data|hypothesis)

    Likelihood(tree,model) = k Prob(observed sequences|tree,model)

    [not Prob(tree|data,model)]


Computing the likelihood of a single tree

C

C

A

G

(1)

(3)

(5)

(2)

(4)

(6)

Computing the likelihood of a single tree

1 jN(1) C…GGACA…C…GTTTA…C(2) C…AGACA…C…CTCTA…C(3) C…GGATA…A…GTTAA…C(4) C…GGATA…G…CCTAG…C


Computing the likelihood of a single tree1

C

C

A

G

C

C

A

G

Prob

+ Prob

A

C

A

A

C

C

A

G

+ … +

Prob

T

T

Computing the likelihood of a single tree

Likelihood at site j =

But use Felsenstein (1981) pruning algorithm


Computing the likelihood of a single tree2

Computing the likelihood of a single tree

Note: PAUP* reports -ln L, so lower -ln L implies higher likelihood


Finding the maximum likelihood tree in principle

Finding the maximum-likelihood tree(in principle)

  • Evaluate the likelihood of each possible tree for a given collection of taxa.

  • Choose the tree topology which maximizes the likelihood over all possible trees.


Probability calculations require

Probability calculations require…

  • An explicit model of substitution that specifies change probabilities for a given branch length“Instantaneous rate matrix”

Jukes-Cantor

Kimura 2-parameter

Hasegawa-Kishino-Yano (HKY)

Felsenstein 1981, 1984

General time-reversible

  • An estimate of optimal branch lengths in units of expected amount of change ( = rate x time)


For example

Jukes-Cantor (1969)

Kimura (1980) “2-parameter”

Hasegawa-Kishino-Yano (1985)

General-Time Reversible

For example:


E g transition probabilities for hky and f84

E.g., transition probabilities forHKY and F84:


A family of reversible substitution models

A Family of Reversible Substitution Models


The relevance of branch lengths

C

C

A

A

A

A

A

A

A

A

C

A

C

C

A

A

A

A

A

A

A

A

C

A

The Relevance of Branch Lengths


When does maximum likelihood work better than parsimony

A

C

B

D

When does maximum likelihood work better than parsimony?

  • When you’re in the “Felsenstein Zone”

(Felsenstein, 1978)


In the felsenstein zone

A

B

0.8

0.8

0.1

0.1

0.1

C

D

A

C

G

T

A

-

5

6

2

C

5

-

3

8

Substitution rates:

G

6

3

-

1

T

2

8

1

-

Base frequencies:

A=0.1

C=0.2

G=0.3

T=0.4

In the Felsenstein Zone


In the felsenstein zone1

In the Felsenstein Zone

1

0.8

parsimony

0.6

Proportion correct

ML-GTR

0.4

0.2

0

0

5000

10000

Sequence Length


Tree searching methods

The long-branch attraction (LBA) problem

Pattern type

14

AI = Uninformative (constant)A

The true phylogeny of

1, 2, 3 and 4

(zero changes required on any tree)

A A

23


Tree searching methods

The long-branch attraction (LBA) problem

Pattern type

14

AI = Uninformative (constant)A

AII = UninformativeG

The true phylogeny of

1, 2, 3 and 4

(one change required on any tree)

A A

23


Tree searching methods

The long-branch attraction (LBA) problem

Pattern type

14

AI = Uninformative (constant)A

AII = UninformativeG

CIII = UninformativeG

The true phylogeny of

1, 2, 3 and 4

(two changes required on any tree)

A A

23


Tree searching methods

The long-branch attraction (LBA) problem

Pattern type

14

AI = Uninformative (constant)A

AII = UninformativeG

CIII = UninformativeG

G IV = MisinformativeG

The true phylogeny of

1, 2, 3 and 4

(two changes required on true tree)

A A

23


Tree searching methods

The long-branch attraction (LBA) problem

… but this tree needs only one step

G

1

A

2

A

3

G

4


Concerns about statistical properties and suitability of models assumptions

Concerns about statistical properties and suitability of models (assumptions)

Consistency

If an estimator converges to the true value of a parameter as the amount of data increases toward infinity, the estimator is consistent.


When do both methods fail

1

3

2

4

When do both methods fail?

  • When there is insufficient phylogenetic signal...


When does parsimony work better than maximum likelihood

When does parsimony work “better” than maximum likelihood?

  • When you’re in the Inverse-Felsenstein (“Farris”) zone

A

C

(Siddall, 1998)

D

B


Siddall 1998 parameter space

Siddall (1998) parameter space

a

b

b

a

b

0.75

p

a

Both methods do poorly

Parsimony has higher

accuracy than likelihood

0

0.75

p

b

Both methods do well


Parsimony vs likelihood in the inverse felsenstein zone

Parsimony vs. likelihood in the Inverse-Felsenstein Zone

1

B

B

B

B

B

B

B

B

B

B

B

0.9

J

15%

B

67.5%

0.8

J

0.7

67.5%

0.6

J

(expected differences/site)

J

Accuracy

0.5

J

J

J

J

0.4

J

J

J

J

0.3

0.2

B

Parsimony

J

ML/JC

0.1

0

20

100

1,000

10,000

100,000

Sequence length


Why does parsimony do so well in the inverse felsenstein zone

Why does parsimony do so well in theInverse-Felsenstein zone?

C

A

C

A

True synapomorphy

A

C

C

C

A

A

A

C

A

Apparent synapomorphies

actually due to

misinterpreted homoplasy

C

C

C

A

A

A

A

G

G

C

C


Parsimony vs likelihood in the felsenstein zone

Parsimony vs. likelihood in the Felsenstein Zone

1

J

J

J

J

J

0.9

J

0.8

J

0.7

67.5%

67.5%

J

0.6

15%

Accuracy

0.5

J

(expected differences/site)

J

0.4

J

0.3

J

0.2

B

Parsimony

B

J

ML/JC

0.1

B

B

0

B

B

B

B

B

B

B

B

B

20

100

1,000

10,000

100,000

Sequence length


From the farris zone to the felsenstein zone

From the Farris Zone to the Felsenstein Zone

A

A

A

C

C

C

D

D

D

B

B

B

A

A

C

C

D

D

B

B

External branches = 0.5 or 0.05 substitutions/site, Jukes-Cantor model of nucleotide substitution


Tree searching methods

G

H

H

H

H

H

G

1.0

H

G

G

G

G

J

J

J

J

J

J

0.8

0.6

Accuracy

0.4

100 sites

J

1,000 sites

G

J

J

10,000 sites

H

0.2

J

J

J

J

G

G

H

H

G

H

G

H

G

G

H

0

H

0.05

0.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

0.05

Farris zon

e

Length of internal branch (

d

)

Felsenstein zone

H

H

1.0

H

H

H

H

G

G

H

H

G

H

G

G

G

0.8

H

J

G

G

J

J

G

0.6

J

J

J

Accuracy

J

G

J

J

0.4

J

H

G

J

G

J

H

100 sites

J

0.2

1,000 sites

G

10,000 sites

H

ML/JC

0

0.05

0.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

0.05

Farris zon

e

Length of internal branch (

d

)

Felsenstein zone

Simulation

results:

Parsimony

Likelihood


Tree searching methods

Maximum likelihood models are oversimplifications of reality. If I assume the wrong model, won’t my results be meaningless?

  • Not necessarily (maximum likelihood is pretty robust)


Model used for simulation

A

B

0.8

0.8

0.1

0.1

0.1

C

D

A

C

G

T

A

-

5

6

2

C

5

-

3

8

Substitution rates:

G

6

3

-

1

T

2

8

1

-

Base frequencies:

A=0.1

C=0.2

G=0.3

T=0.4

Model used for simulation...


Performance of ml when its model is violated one example

Performance of ML when its model is violated (one example)


Among site rate heterogeneity

Among site rate heterogeneity

equal rates?

Lemur AAGCTTCATAG TTGCATCATCCA …TTACATCATCCA

Homo AAGCTTCACCG TTGCATCATCCA …TTACATCCTCAT

Pan AAGCTTCACCG TTACGCCATCCA …TTACATCCTCAT

Goril AAGCTTCACCG TTACGCCATCCA …CCCACGGACTTA

Pongo AAGCTTCACCG TTACGCCATCCT …GCAACCACCCTC

Hylo AAGCTTTACAG TTACATTATCCG …TGCAACCGTCCT

Maca AAGCTTTTCCG TTACATTATCCG …CGCAACCATCCT

  • Proportion of invariable sites

    • Some sites don’t change do to strong functional or structural constraint (Hasegawa et al., 1985)

  • Site-specific rates

    • Different relative rates assumed for pre-assigned subsets of sites

  • Gamma-distributed rates

    • Rate variation assumed to follow a gamma distribution with shape parameter 


Performance of ml when its model is violated another example

.

.

.

.

.

Performance of ML when its model is violated (another example)

Modeling among-site rate variation with a gamma distribution...

0.08

a=200

0.06

a=0.5

a=2

Frequency

0.04

a=50

0.02

0

0

1

2

Rate

…can also estimate a proportion of “invariable” sites (pinv)


Performance of ml when its model is violated another example1

Performance of ML when its model is violated (another example)


Moderate felsenstein zone

“MODERATE”–Felsenstein zone


Moderate inverse felsenstein zone

“MODERATE”–Inverse-Felsenstein zone


Bayesian inference in phylogenetics

Bayesian Inference in Phylogenetics

  • Uses Bayes formula:

    Pr(|D) = Pr(D|) Pr() Pr(D)

     Pr(D|) Pr()

     L() Pr()

  • Calculation involves integrating over all tree topologies and model-parameter values, subject to assumed prior distribution on parameters

( =tree topology, branch-lengths, and substitution-model parameters)


Bayesian inference in phylogenetics1

Bayesian Inference in Phylogenetics

  • To approximate this posterior density (complicated multidimensional integral) we use Markov chain Monte Carlo (MCMC)

    • Simulated Markov chain in which transition probabilities are assigned such that the stationary distribution of the chain is the posterior density of interest

    • E.g., Metropolis-Hastings algorithm: Accept a proposed move from one state  to another state * with probability min(r,1) where

      r =Pr(*|D) Pr(| *)

      Pr(|D) Pr(*| )

    • Sample chain at regular intervals to approximate posterior distribution

  • MrBayes (by John Huelsenbeck and Fredrik Ronquist) is most popular Bayesian inference program


Tree searching methods

B

A

“burn in”

C

D

B

A

B

B

B

A

A

A

C

D

B

A

A

D

D

D

A

C

C

C

A

C

B

C

D

C

D

B

D

D

B

C

C

A

A

B

AB|CD

AB|CD

AB|CD

BC|AD

BC|AD

BC|AD

BC|AD

AC|BD

D

C

B

D

Initialize the chain, e.g., by picking a random state X0 (topology,branch lengths, substitution-model parameters) from the assumed prior distribution

2.For each time t, sample a new candidate state Y from some proposal distribution q(.|Xt) (e.g., change branch lengths or topology plus branch lengths)

Calculate acceptance probability

A brief intro to Markov chain Monte Carlo (MCMC)

...

Likelihood

Iterations

If Y is accepted, let Xt+1 = Y; otherwise let Xt+1 = Xt

If the chain is run “long enough”, the stationary distribution of states in the chain will represent a good approximation to the target distribution (in this case, the Bayesian posterior)


Model based distances

Model-based distances

  • Can also calculate pairwise distances based on these models

  • These distances estimate the number of substitutions per site that have accumulated since the two sequences shared a common ancestor, allowing for superimposed substitutions (“multiple hits”)

  • E.g.:

    • Jukes-Cantor distance

    • Kimura 2-parameter distance

    • General maximum-likelihood distances available for other models


Distance based optimality criteria additive trees

1

3

1

2

a

d

c

3

b

e

4

2

4

1

2

3

4

Distance-based optimality criteria“Additive trees”

p12 = a+b

p13 = a+c+d

p14 = a+c+e

p23= b+c+d

p24= b+c+e

p34= d+e

pij = dij for all i and j if the tree

topology is correct and distances

are additive


Distance based optimality criteria minimum evolution and least squares

Distance-based optimality criteriaMinimum evolution and least-squares

Minumum

evolution

(ME)

Least-Squares

pij

dij

SS

LS branch lengths


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