Introduction to Phylogenetics - Barbara Holland

Introduction to Phylogenetics

Barbara Holland

BioInfoSummer 2013

What is phylogenetics?

Darwin’s sketch: the first phylogenetic tree?

The goal of phylogenetics is to infer evolutionary relationships between species.

This includes both information about order of branching, .e.g., did humans and chimpanzees share a common ancestor more recently than humans, chimps and gorillas? And information about timing of events, e.g., how long ago did humans and chimps share an ancestor?

Since the publication of Origin of the Species in 1859 people have been trying to infer the evolutionary “Tree of Life”.

Ernst Haeckel’s Tree of Life (1866)

Why might we care?

• Understanding human origins

• Understanding biogeography, e.g. what’s the relative importance of dispersal versus vicariance?

• Understanding the origin of particular traits

• Understanding the processes of molecular evolution

• Learning about the tempo of evolution, e.g. was the Cambrian explosion really an explosion? Did mammals and birds wait until dinosaurs went extinct to inherit the earth or were they already started before the asteroid hit?

• Origin of disease, e.g. where did humans get AIDs from?

Understanding the emergence of disease

http://evolution.berkeley.edu/evolibrary/news/081101_hivorigins

Special reasons for Australians to care…

http://archive.peabody.yale.edu/exhibits/treeoflife/predictions.html

How to “read” a tree

Rat

Mouse

Cat

Dog

A tree is a connected acyclic graph

Composed of:Nodes / verticesEdges / arcs / branchesPendent (external) edges versus internal edges

In a phylogenetic tree the external (degree 1) nodes are associated with labels

• Trees can be rooted or unrooted• Rooted trees are more interpretable• But, most software returns unrooted trees

Tree basics

Cat Dog MouseRat

root Rat

Mouse

Cat

Dog

• Trees can be weighted or unweighted

• Edge weights are used to represent – the amount of genetic change along the edge– time

CatDog

Mouse

Rat

Rat

Mouse

Cat

Dog

Tree basics

• Trees can be bifurcating (binary) or multifurcating (non-binary)

• Polytomies usually represent uncertainty (a soft polytomy), but sometimes they are used to mean that everything happened at exactly the same time (a hard polytomy).

Human Chimp Gorilla Monkey

polytomy(multifurcation)

Tree basics

The same tree can look different

A AB BC D CDE F F E

Newick format

A B C D E F

A computer readable format for describing trees that uses brackets and commas.

((A,(B,(C,D))),(E,F));

( , );((A, ),(E,F));((A,(B, )),(E,F));

Treeview / Dendroscope

What data can we use?• Molecular sequence data

– DNA alignments*– Amino acid alignments

• Presence absence data– Gene content– Fragment based methods (AFLP, DArT)– SNP chips

• Genetic distances (DNA-DNA hybridisation, immunology)• Rare traits

– Gene order– Introns?– SINEs and LINEs (short and long interspersed retro-transposable elements )

• Morphological data

* Alignment is an important problem in its own right

The molecular phylogeny problem

ACCGCTTA

ACTGCTTA

ACTGCTAAACTGCTTA

ACCCCTTA

ACCCCTTA

Tim

e

ACCCCATA

…ACCCCTTA……ACCCCATA……ACTGCTTA……ACTGCTAA…

We see the alignedmodern day sequences

And want to recover theunderlying evolutionarytree.

Sometimes the data agrees

ACCGCTTA

ACTGCTTA

ACTGCTAAACTGCTTA

ACCCCTTA

ACCCCTTA

Tim

e

ACCCCATA

ACCCCTTAACCCCATAACTGCTTAACTGCTAA

Sometimes not

ACCGCTTA

ACTGCTTA

ACTGCTAAACTGCTTC

ACCCCTTA

ACCCCTTC

Tim

e

ACCCCATA

ACCCCTTCACCCCATAACTGCTTCACTGCTAA

How can we choose the best tree?

• Distance data– Quick methods that take a greedy clustering approach (e.g.

Neighbour-Joining)– Optimality criteria that assign each tree a score (Minimum evolution,

fastME)

• Character data (e.g. a sequence alignment)– Non-parametric methods (parsimony)– Model-based methods (maximum likelihood, Bayesian)

Parsimony

To decide which tree is best we can use an optimality criterion. This means we need a way of assigning each possible tree a score.

Maximum Parsimony is one such criterion. It chooses the tree which requires the fewest substitutions to explain the data.

The Principle of Parsimony is the general scientific principle that accepts the simplest of two explanations as preferable.

S1 ACCCCTTC S2 ACCCCATA S3 ACTGCTTC S4 ACTGCTAA(1,2),(3,4)(1,3),(2,4)

1

2

3

4

1

3

2

4

S1 ACCCCTTC S2 ACCCCATA S3 ACTGCTTC S4 ACTGCTAA(1,2),(3,4) 0(1,3),(2,4) 0

A

A

A

A

A

A

A

A


C

C

T

T

C

T

C

T

C T

C

TT

C


C

C

G

G

C

G

C

G

C G

C

GG

C


T

A

T

T

T

T

A

T

T

A

A

T


T

T

T

A

T

T

T

A

T

A

T

A

S1 ACCCCTTC S2 ACCCCATA S3 ACTGCTTC S4 ACTGCTAA(1,2),(3,4) 00110112 6(1,3),(2,4) 00220111 7

C

A

C

A

C

C

A

A

A

C

C

A

C A

According to the parsimony optimality criterion we should prefer the tree (1,2),(3,4) over the tree (1,3),(2,4) as it requires the fewest mutations.

The “large parsimony problem”

The small parsimony problem – to find the score of a given tree - can be solved in linear time in the size of the tree (using the Fitch or Sankoff algorithm).

The large parsimony problem, finding the tree with minimum score, is known to be NP-Hard.

How many trees are there?

#species #unrooted binary tip-labelled trees

4 3

5 3*5=15

6 3*5*7=105

7 3*5*7*9=945

10 2,027,025

20 2.2*1020

n (2n-5)!!

An exact search for the best tree, where each tree is evaluated according to some optimality criterion such as parsimony quickly becomes intractable as the number of species increases

Counting trees

1

2 3

1 1 1

1

2

2 2

5 2

43

3

3 34 44

1 3 2

45

5 1 2

43

1 2 5

43

1 4 2

53

1

1 x 3 = 3

1 x 3 x 5 = 15

Search strategies• Exact search: possible for small numbers of taxa (~12 or less) only

• Branch and Bound: A smarter way of doing exact search, up to ~20 taxa

• Greedy search: sequentially build up the tree with no backtracking (e.g. Neighbour-Joining)

• Local Search – Heuristics: pick a good starting tree and use moves within a “neighbourhood” to find a better tree.

• Meta-heuristics:– Genetic algorithms– Simulated annealing– The ratchet

The “Felsenstein Zone” Parsimony was once a very popular method of phylogenetic inference but

it has fallen from grace in recent years due to the fact that it is statistically inconsistent for many relevant phylogenetic models.

Consistency in statistics is the property that as you get more and more data you are more and more likely to get the correct answer. i.e. in the case of phylogenetics this means that as you get longer and longer sequence alignments then you'd expect to be increasingly likely to infer the tree that these sequences evolved on.

Felsenstein's example• The true tree is ((A,B),C,D)

• The only parsimony informative patterns are of the form: xxyy, xyxy, and xyyx

• Let p be the probability of mutation on a short edge and let q be the probability of mutation on a long edge

• Felsenstein showed that if p was small compared to q then the chance of seeing a parsimony informative pattern that “matched” (i.e. cost 1 on) the true tree, xxyy, was smaller than the chance of seeing a pattern, xyxy, that cost 1 on the incorrect tree ((A,C),B,D)

• For large datasets Parsimony would be guaranteed to get the incorrect tree

A C

B D

A C

B D

True tree

Long branch attraction tree

Inconsistency with a clock

Felsenstein’s 4 taxon example appeared in a 1978 paper “Cases in which parsimony or compatibility methods will be positively misleading.” Syst. Zool.

The “misleading zone” was conjectured to depend on the highly non-clocklike nature of Felsenstein’s tree.

In 1989 Hendy and Penny showed that clocklike trees could also suffer from the same effect.

Outgroup1 2 3 4

The 4 trees ((1,out),(3,4),2) and ((2,out),(3,4),1) and ((3,out),(1,2),4) and ((4,out),(1,2),3) all have lower expected parsimony score than the true tree ((1,2),(3,4),out)

Enter statistical phylogenetics

• Over the last 30 years model-based methods of phylogenetic inference have come to the fore.

• These methods assume a model of nucleotide substitution which specifies the rates that nucleotides (A,C,G,T) mutate into other nucleotides.

• They aim to find the tree with the highest probability of generating the sequence data observed, i.e. the maximum likelihood tree

Sequence evolution is modelled as a Markov process

time

A

C

Consider a single edge in a phylogeny, i.e. evolution of a single species, and the evolution of a single DNA base amongst the possible states {A, C, G, T}.

The probability of mutating from state i to j over a length of time t depends only on the current state i and the potential future state j, not on any of the previous history of the sequence, and can be written pij(t).

A

T

G

time

t

Continuous time Markov chains

A C G T

A pAA pAC pAG pAT

C pCA pCC pCG pCT

G pGA pGC pGG pGT

T pTA pTC pTG pTT

M =

A C G T

A -qA* qAC qAG qAT

C qCA -qC* qCG qCT

G qGA qGC -qG* qGT

T qTA qTC qTG -qT*

Q = Where qi* = Σj qij, j ≠ i

i.e. rows sum to zero.

M = exp(Qt)

Transition matrix Instantaneous rate matrix

Typically we restrict to stationary, reversible models, with the stationary distribution denoted by π. So, π Q = 0, and D(π)Q is symmetric.

Example with a simple model

2-state symmetric model (R := AG purines, Y := CT pyrimidines)

Each edge i has probability pi of a mutation determined by its length

a …R…b …R…c …Y…d …Y…

a

b

c

d

0.1

0.1

0.20.15

0.1

R Yα

Example with a simple model

a …R…b …R…c …Y…d …Y…

R

R

Y

Y

0.1

0.1

0.20.15

0.1

L = 0.9*0.9*0.8*0.15*0.1 + 0.9*0.9*0.2*0.85*0.9 + 0.1*0.1*0.2*0.15*0.1 + 0.1*0.1*0.8*0.85*0.9

R

R

Y

YR

R

R

Y

Y

R

R

Y

Y

R

R

Y

Y

? ?

RR RY Y Y Y

In theory you can then…

• Multiply the likelihoods over all sites to get the likelihood of a particular weighted tree (in practice you sum the log likelihoods).

• Optimise the edge lengths to find the maximum likelihood value the tree.

• Compare the maximum likelihood score of all trees (in practice you would probably use heuristic search).

• Report the weighted tree with the highest overall likelihood.

Computational complexity

• Maximum likelihood is computationally complex.

• Fortunately the “tiny problem”, finding the likelihood of a site on a tree with fixed branch lengths, is solvable in linear time using Felsenstein’s pruning algorithm (dynamic programming)

• However, solving the “small problem”, i.e. computing the likelihood of a particular tree, is time consuming. Unlike parsimony, here the edge lengths matter and must be optimized for each tree.

• Finding the best edge weights for a given tree uses local search (hill-climbing), and can get stuck in local optima.

Models of nucleotide substitution

• Jukes Cantor (JC)– All substitutions equally likely– Base frequencies equal

• Kimura 2 Parameter (K2P)– Transitions and transversions at

different rates– Base frequencies equal

• HKY model– Transitions and transversions at different rates– Base frequencies different

• General Time Reversible (GTR)

A G

C T

α

α

αα

α

α

Models of nucleotide substitution• Jukes Cantor (JC)

– All substitutions equally likely– Base frequencies equal





A G

C T

β

αα

α

α

β







A G

C T

β

αα

α

α

β







A G

C T

α

ζ

εγ

β

δ

Extra features that can be modelled

• Site to site rate variation (usually modelled by a gamma distribution)• Invariant sites• Different parameters for different genes/codon positons

BUT• Some parts of reality are problematic…

– Base composition bias (LogDet)– Sites that are free to vary change across the tree– Non independence of sites

Stochastic versus Systematic error

• Rather than just getting a point estimate of the tree we are usually interested in getting some measure of confidence

• Error could arise in two ways– Stochastic – our sequences are too short– Systematic – our models are wrong

• There are good methods for determining whether or not stochastic error is a problem.

Assessing confidence in trees • We would like some measure of confidence in the inferred tree.

– Is the tree likely to change if we got more data, or if we had used slightly different data?

– Are some parts of the tree more robust than others?

• The bootstrap is a useful tool for answering these sorts of questions.

The bootstrap

• In 1985 Felsenstein introduced the idea of the bootstrap to phylogenetics.

• For each bootstrap sample– Create a new alignment by resampling the columns of the observed

alignment– Construct a tree for the ‘bootstrap’ alignment

• Can be applied to any method that starts from a sequence alignment, e.g., parsimony, likelihood, clustering methods if the distances are derived from an alignment…

• The bootstrap support for each edge is the number of bootstrap trees that edge appears in.

1234567a ATATAAAb ATTATAAc TAAAATAd TATAAAT

1224567a ATTTAAAb ATTATAAc TAAAATAd TAAAAAT

1334567a AAATAAAb ATTATAAc TAAAATAd TTTAAAT

1234567a ATATAAAb ATTATAAc TAAAATAd TATAAAT

1244567a ATTTAAAb ATAATAAc TAAAATAd TAAAAAT

a a a a

b

b

b b

c

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c c

d d d d

a

b

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d

0.75

Example where the bootstrap is useful

• Simulate data on the four taxon tree below (JC model)

• Use sequence lengths of 100, 1000, and 10000

100 1000 10000((a,b),(c,d)) 5.7% 97% 100%((a,c),(b,d)) 42.8% <5% 0((a,d),(b,c)) 49.8% <5% 0

0.2

0.01

a b c d

Example where the bootstrap is not so useful

• Simulate data on the two four-taxon trees below (JC model) in the proportion 55%, 45% and concatenate the sequences

• Use total sequence lengths of 100, 1000, and 10000

100 1000 10000((a,b),(c,d)) 64% 80% 98%((a,c),(b,d)) 33% 20% <5((a,d),(b,c)) 3% 0% <5

0.1

0.05

a b c d

0.1

0.05

a c b d

55%

45%

Mistaking precision for accuracy106 nuclear genes: Different methods provide conflicting Yeast topologies, each with 100% bootstrap support

The results show the importance of systematic error

Phillips et al. (MBE, 2004)

Phylogenomics

• With increasing quantities of sequence data it is common to get very well resolved trees, i.e. bootstrap values (or posterior probabilities) close to 100% (or 1)

• HOWEVER, slight changes to the model or inference method used can mean you get 100% support for different trees?!

• This suggests that it is very important to know how well our models fit our data.

Testing model fit

• In phylogenetic studies we often ask questions of the form “Is model A better than model B?” (relative goodness-of-fit tests, based on likelihood ratio tests or information criteria such as the AIC or BIC)

• We less frequently ask questions of the form “Is model C a good fit to the data?” (absolute goodness-of-fit tests)

• The alst question is clearly important as if our models fit poorly our results may be subject to systematic errors

• So why is the last question less popular?

Absolute GOF tests

• Absolute tests of goodness-of-fit are fiddlier to implement than relative tests. They typically require doing many parametric simulations under the best model and then summarizing each simulated dataset by a single test statistic.

• With very large datasets you almost always will be able to reject your model.

• If the best model fails an absolute GOF test, it’s not clear what you should do next…

Health & Medicine

Introduction to Phylogenetics - Barbara Holland