Maximum likelihood methods of phylogenetic inference evaluate a hypothesis about evolutionary history (the branching order and branch lengths of a tree) in terms of a probability that a proposed model of the evolutionary process and the hypothesised histo

Maximum Likelihood 1

Maximum likelihood estimates a parameter from observed data under an explicit model

There is an explicit link between model + tree + data (poor model = poor tree?)
- Likelihood also provides ways of evaluating models in terms of their log likelihoods, provided they are nested i.e. one model is a special case of the other
- Different trees can also be evaluated for their fit to the data under a particular model (likelihood difference tests of two trees after Kishino & Hasegawa)

Maximum Likelihood 2

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1 CGAGAC

2 AGCGAC

3 AGATTA

4 GGATAG

What is the probability that unrooted Tree A (rather than another tree) could have generated the data shown under our chosen model ?

Maximum likelihood tree reconstruction 1

Tree A

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1 CGAGAC

2 AGCGAC

3 AGATTA

4 GGATAG

What is the probability that unrooted Tree A (rather than another tree) could have generated the data shown under our chosen model ?

Maximum likelihood tree reconstruction 1

Tree A

note rooting is arbitrary

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1 CGAGA C

2 AGCGA C

3 AGATT A

4 GGATA G

ACGT

The likelihood for a particular site j is the sum of the probabilities of every possible reconstruction of ancestral states under a chosen model

Maximum likelihood tree reconstruction 2

4 x 4 possibilities

Tree A

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Maximum likelihood tree reconstruction 3

The likelihood of Tree A is the product of the likelihoods at each site

The likelihood is usually evaluated by summing the log of the likelihoods (because the summed probabilities are so small) at each site and reported as the log likelihood of the full tree

The Maximum likelihood tree is the one with the highest likelihood (might not be Tree A i.e. it could be another tree topology)

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Maximum likelihood tree reconstruction 4

How are the probabilities of change calculated ?

The probabilities used to calculate likelihoods depend on the assumed model

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The probability of any change is independent of the prior history of the site (a Markov Model)

Substitution probabilities do not change with time or over the tree (a homogeneous Markov process)

Change is time reversible e.g. the rate of change of A to T is the same as T to A

Typical assumptions of ML substitution models

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The model incorporates information about the rates at which each nucleotide is replaced by each alternative nucleotide
- For DNA this can be expressed as a 4 x 4 rate matrix

Other model parameters may include:
- Site by site rate variation - often modelled as a statistical distribution - for example a gamma distribution

Maximum likelihood models 1

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Model parameters can be:
- estimated from the data (using maximum likelihood in PAUP)
- can be pre-set based upon assumptions about the data (for example that for all sequences all sites change at the same rate and all substitutions are equally likely - e.g. the widely used Jukes and Cantor Model)
- wherever possible avoid assumptions which are violated by the data because they can lead to incorrect trees

Maximum likelihood models 2

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A case study in phylogenetic analysis:

Deinococcus and Thermus

Deinococcus are radiation resistant bacteria

Thermus are thermophilic bacteria
Both have the same very unusual cell wall based upon ornithine
Both have the same menaquinones (Mk 9)
Both have the same unusual polar lipids

Congruence between these complex characters supports a phylogenetic relationship between Deinococcus and Thermus

A four taxon problem for Deinococcus and Thermus(Thermus, Deinococcus, Bacillus, Aquifex)

Aquifex and Bacillus are thermophiles and mesophiles, respectively

No data suggest that Aquifex and Bacillus are specifically related to either Deinococcus or Thermus

If all four bacteria are included in an analysis the true tree should place Thermus and Deinococcus together

Thermus

Deinococcus

Aquifex

Bacillus

“The true tree”

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The Jukes and Cantor model is the simplest model

The JC model is a one parameter model

1) it assumes that all changes are equally probable (p=0.25)

2) unless modified it assumes all sites can change and that they do so at the same rate

Output of JC ML analysis for (Thermus, Deinococcus, Bacillus, Aquifex)

Tree 1 -log likelihood = 4090

Best tree

Tree 2 -log likelihood = 4101

True tree

Tree 3 -log likelihood = 4132

The Jukes and Cantor model in ML is unable to recover the true tree for this data set

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The 16S rRNA genes of Aquifex, Bacillus, Deinococcus and Thermus

Exclude characters command in PAUP - exclude constant sites:

Character-exclusion status changed:

859 of 1273 characters excluded

Total number of characters now excluded = 859

Number of included characters = 414

Taxon A C G T # sites

--------------------------------------------------------------

Aquifex 0.12319 0.38164 0.38164 0.11353 414

Deinococc 0.23188 0.22222 0.27295 0.27295 414

Thermus 0.13317 0.35835 0.37530 0.13317 413

Bacillus 0.23188 0.22705 0.26570 0.27536 414

--------------------------------------------------------------

Mean 0.18006 0.29728 0.32387 0.19879 413.75

Base frequencies command in PAUP:

Does the JC model fit these data?

Models can be made more parameter rich to increase their realism 1

The most common additional parameters are:
- A correction to allow different substitution rates for each type of nucleotide change
- A correction for the proportion of sites which are unable to change
- A correction for variable site rates at those sites which can change

PAUP will estimate the values of these additional parameters for you

A gamma distribution can be used to model site rate heterogeneity

Models can be made more parameter rich to increase their realism 2

But the more parameters you estimate from the data the more time needed for an analysis and the more sampling error accumulates
- One might have a realistic model but large sampling errors
- Realism comes at a cost in time and precision!
- Fewer parameters may give an inaccurate estimate, but more parameters decrease the precision of the estimate
- In general use the simplest model which fits the data

Use PAUP to compare nested models incorporating additional parameters for their likelihoods

Models can be made more parameter rich to increase their realism 3

JC ML tree

-4090

JC -invariable sites - 4030

JC -inv + gamma

correction for variable sites - 4029

GTR-inv + gamma

correction for variable sites - 3985

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The GTR model of sequence evolution:

The general time reversable model (GTR) is the most general substitution model because it assigns different rates for each type of substitution. For example for the 16S ribosomal RNA data for Deinococcus, Thermus, Aquifex and Bacillus:

Tree number 1:

-Ln likelihood = 3985.30400

Estimated R-matrix:

-2.7325625 0.4419956 1.42028 0.87028688

0.4419956 -5.2448524 1.2621698 3.540687

1.42028 1.2621698 -3.6824498 1

0.87028688 3.540687 1 -5.4109739

Estimated value of proportion of invariable sites = 0.228318

Estimated value of gamma shape parameter = 0.610459

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The 16S rRNA genes of Aquifex, Bacillus, Deinococcus, Thermus and Thermus ruber

Exclude characters command in PAUP - exclude constant sites:

Base frequencies command in PAUP:

Character-exclusion status changed:

837 characters excluded

Total number of characters now excluded = 837

Number of included characters = 436

Taxon A C G T # sites

--------------------------------------------------------------

ruber 0.19725 0.27294 0.29587 0.23394 436

Aquifex 0.12156 0.38073 0.38532 0.11239 436

Deinococc 0.22477 0.22936 0.28211 0.26376 436

Thermus 0.13103 0.35862 0.37931 0.13103 435

Bacillus 0.22477 0.23394 0.27523 0.26606 436

--------------------------------------------------------------

Mean 0.17990 0.29509 0.32354 0.20147 435.80

Output of GTR-inv sites ML analysis for (Deinococcus, Bacillus, Aquifex, thermus and Thermus ruber)

Tree 1 -log likelihood = 4439

Tree 2 -log likelihood = 4447

Tree 3 -log likelihood = 4437

Best tree = True tree

With the addition of Thermus ruber which has a base composition which is intermediate between thermophiles and mesophiles GTR-inv sites ML recovers the Thermus + Deinococcus relationship

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Yang (1995) has shown that parameter estimates are reasonably stable across tree topologies provided trees are not “too wrong”. Thus one can obtain a tree using a quick method (useful when many sequences are being analysed) and then estimate parameters

Estimation of ML substitution model parameters:

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Parameter estimates using the “tree scores” command in PAUP*

Use PAUP* tree scores to use ML to estimate over this tree:

1) Proportion of invariant sites

2) Gamma shape parameter for variable sites

3) Substitution parameters for all types of change

Maximum parsimony tree

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ML Parameter estimates over a parsimony tree using tree scores in PAUP*

Tree number 1:

-Ln likelihood = 4432.16903

Estimated R-matrix:

-2.992539 0.53399075 1.6835489 0.77499941

0.53399075 -6.0877637 1.0048052 4.5489678

1.6835489 1.0048052 -3.6883541 1

0.77499941 4.5489678 1 -6.3239672

Corresponding Q-matrix:

-0.77509276 0.12637319 0.52569668 0.12302289

0.11475088 -1.1506065 0.31375553 0.72210013

0.36178289 0.2377952 -0.75831742 0.15873934

0.16654196 1.0765496 0.31225508 -1.5553467

Estimated value of proportion of invariable sites = 0.302946

Estimated value of gamma shape parameter = 0.629797

These values can then be used as the starting parameters for a full likelihood search

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Maximum Likelihood Tree

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Mathematically rigorous & performs well in computer simulations

Allows investigation of the fit between model and data

Provides a simple way of comparing trees according to their likelihoods (difference tests - Kishino Hasegawa Test)

Maximum Likelihood -advantages

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Maximum likelihood will only be consistent (converge on the true tree) if evolution proceeds according to the assumed model: How well does the model fit the data ?

Becomes impossible computationally if many taxa or many model parameters

Maximum Likelihood -disadvantages