Phylogenetics

“Inferring Phylogenies”

Joseph Felsenstein

Excellent reference

What is a phylogeny?

Different Representations Cladogram - branching pattern only Phylogram - branch lengths are

estimated and drawn proportional to the amount of change along the branch

Rooted - implies directionality of change Unrooted - does not How do you root a tree?

What is a phylogeny used for?

€

π =2 π ij

j=i+1

n

∑i=1

n−1

∑

n n−1( )

€

θ =4N eiμ

Estimate a Phylogeny

Sp1 ACCGTCTTGTTASp2 AGCGTCATCAAASp3 AGCGTCATCAAASp4 ACCGTCTTGATASp5 AGCCTCTTCATA

Estimate a Phylogeny

Sp1 ACCGTCTTGTTASp2 AGCGTCATCAAASp3 AGCGTCATCAAASp4 ACCGTCTTGATASp5 AGCCTCTTCATA

Working Tree

sp1

sp4

sp2

sp3

sp5

c2

Estimate a Phylogeny

Sp1 ACCGTCTTGTTASp2 AGCGTCATCAAASp3 AGCGTCATCAAASp4 ACCGTCTTGATASp5 AGCCTCTTCATA

Working Tree

sp1

sp4

sp2

sp3

sp5

c2

c4

Estimate a Phylogeny

Sp1 ACCGTCTTGTTASp2 AGCGTCATCAAASp3 AGCGTCATCAAASp4 ACCGTCTTGATASp5 AGCCTCTTCATA

Working Tree

sp1

sp4

sp2

sp3

sp5

c2

c4

c7

Estimate a Phylogeny

Sp1 ACCGTCTTGTTASp2 AGCGTCATCAAASp3 AGCGTCATCAAASp4 ACCGTCTTGATASp5 AGCCTCTTCATA

Working Tree

sp1

sp4

sp2

sp3

sp5

c2

c4

c7

c9

Estimate a Phylogeny

Sp1 ACCGTCTTGTTASp2 AGCGTCATCAAASp3 AGCGTCATCAAASp4 ACCGTCTTGATASp5 AGCCTCTTCATA

Working Tree

sp1

sp4

sp2

sp3

sp5

c2

c4

c7

c9

c10

Estimate a Phylogeny

Sp1 ACCGTCTTGTTASp2 AGCGTCATCAAASp3 AGCGTCATCAAASp4 ACCGTCTTGATASp5 AGCCTCTTCATA

Final Tree

sp1

sp4

sp2

sp3

sp5

c2

c4

c7

c9

c10 c11

What optimality criteria do we use then? Parsimony Likelihood Bayesian

Distance methods?

Parsimony Why should we choose a specific grouping? Maximum parsimony: we should accept the

hypothesis that explain the data most simply and efficiently

“Parsimony is simply the most robust criterion for choosing between competing scientific hypotheses. It is not a statement about how evolution may or may not have taken place”1

1 Kitching, I. J.; Forey, P. L.; Humphries, J. & Williams, D. M. 1998. Cladistics: the theory and practice of parsimony analysis. The systematics Association Publication. No. 11.

Parsimony Optimality criteria that chooses the

topology with the less number of transformations of character states

Optimizing one component: tree topology (pattern based)

Most parsimonious tree: the one (or multiple) with the minimum number of evolutionary changes (smaller size/tree length)

Reconstructing trees via sequence data1 2 3 4 5 6

O T G T A A T

A A A T G A G

B A G C C - G

C A A T G A T

D A G C C - T

AO DC B

1. T=>A

3. T=>C

2. G=>A

4. A=>G

4. A=>C5. A=> GAP

6. T=>G6. T=>G

Tree length = 8

Neighbor-joining Method

NJ distance matrices

NJ distance matrices

NJ distance matrices

NJ distance matrices

Finished NJ tree

Pyrimidines

Purines

T C

A G

Models of Evolution

Transversions Transitions

Maximum Likelihood

Base frequencies: fA + fG + fC + fT = 1 Base exchange: fs + fv = 1 R-matrix: + + + + + = 1 Gamma shape parameter Number of discrete gamma-distribution categories Pinvar: fvar + finv = 1

Likelihood: L = li where i is each character state

Maximum Likelihood

L=Pr(D|H)

L

( i )

= Pr AGGCG via x , y , z , w( )

all x , y , z , w

∑

= (Pr w ) (Pr z ; w , t8

) (Pr x ; w , t7

) (Pr y ; z , t6

) (Pr A ; x , t1

) (Pr G ; z , t2

) (Pr G ; z , t3

) (Pr C ; y , t4

) (Pr G ; y , t5

)

w

∑

z

∑

y

∑

x

∑

w

zx

y

GC

GGA

t1 t2 t3

t4 t5

t6

t7 t8

ML cont.

L = L

( i )

i = 1

n

∏

Pii

( t ) =

1

4

+

3

4

e

− 4 λ t / 3

the probability that the nucleotide at time t is i is given by

the probability that the nucleotide at time t is j, j i, is given by

Pij

( t ) =

1

4

−

1

4

e

− 4 λ t / 3

Bayes Theorem

Prob (H │D) = Prob (H) Prob (D│H)

Prob (D)H=Hypothesis D=Data

Prior probability orMarginal probability of HThe conditional

probability of H given D: posterior probability

Likelihood function

Prior probability orMarginal probability of D∑HP(H) P(D|H)

Normalizing Constant: ensures ∑ P (H │D) = 1

Take Home Message Likelihood: represents the P of the data

given the hypothesis => difficult to interpret

Bayes approach: estimates the P of the hypothesis given the data => estimates P for the hypothesis of interest

Bayesian Inference of Phylogeny

Calculating pP of a tree involves a summation over all possible trees and, for each tree, integration over all combinations of bl and substitution-model parameter values

f(i |X) = f(i) f(X|i)∑j=1 f(i) f(X|i)

B(s)

f(i,i,|X) = f(i,i,) f(X|i,i,)∑j=1 ∫ , f(i,i,) f(X| i,i,)dd

B(s)

f(i|X) = ∫ , f(i,i,) f(X|i,i,) dd∑j=1 ∫ , f(i,i,) f(X| i,i,)ddB(s)

Inferences of any single parameter are based on the marginal distribution of the parameter

This marginal P distribution of the topology, for example, integrates out all the other parameters

Advantage: the power of the analysis is focused on the parameter of interest (i.e., the topology of the tree)

Estimating phylogenies Exhaustive Searches Branch and bound methods Rise in computational time versus rise

in solution space

How many topologies are there?

€

T =2n − 3( )!

2n−1 n −1( )!

The Phylogenetic Problem

Number of Seqs Number of Trees10 2x106

100 2x10182

1,000 2x102,860

10,000 8x1038,658

100,000 1x10486,663

1,000,000 1x105,866,723

B(T)= 2i−5( )i=3

T∏

HIV-1 Whole Genomes1993 - 15

HIV-1 Whole Genomes2003 (JAN) - 397

Tree Space - the final frontier

Heuristic Searches Nearest-neighbor interchanges (NNI) - swap two adjacent

branches on the tree Subtree pruning and regrafting (SPR) - removing a branch

from the tree (either an interior or an exterior branch) with a subtree attached to it. The subtree is then reinserted into the remaining tree in all possible places

Tree bisection and reconnection (TBR) - An interior branch is broken, and the two resulting fragments o the tree ar considered as separate trees. All possible connections are made between a branch of one and a branch of the other.

Other approaches Tree-fusing - find two near optimal trees

and exchange subgroups between the two trees

Genetic Algorithms - a simulation of evolution with a genotype that describes the tree and a fitness function that reflects the optimality of the tree

Disc Covering - upcoming paper

Phylogenetic Accuracy? Consistency - A phylogenetic method is consistent for a given evolutionary model if the method converges on the correct tree as the data available to the method become infinite.

Efficiency - Statistical efficiency is a measure of how quickly a method converges on the correct solution as more data are applied to the problem.

Robustness - Robustness refers to the degree to which violations of assumptions will affect performance of phylogenetic methods

How reliable is MY phylogeny? Bootstrap Analysis Jackknife Analysis Posterior Probabilities (Bayesian

Approaches) Decay Indices

Bootstrap

Pseudoreplicates