1. Probabilistic methods for phylogenetic trees (Part 2)
Sushmita Roy
BMI/CS 576
Oct 7th, 2014

2. Probabilistic methods for phylogenetic tree construction
RECAP
Probabilistic methods for phylogenetic tree construction
P(data|tree)
Maximum likelihood
Felsenstein algorithm for computing the likelihood of a sequence given a tree

3. Probabilistic models of evolution
The probability of a character switching from a to b along a branch of length t, P(b|a,t) is captured by the matrix
For example for DNA this is:

4. Defining the conditional probability distributions
If we consider t to be evolutionary time, these conditional probabilities can be obtained what is called a continuous time Markov process
Such processes are defined by a K-by-K rate matrix R
Each entry of R, R(a,b) gives a rate of substitution from a to b
The time spent in any state (character) is exponentially distributed
If we have R, S(t) can be obtained from R
Using the theory of continuous time Markov processes

5. Rate matrices A rate matrix R
Is a K-by-K matrix where K is the size of our alphabet
E.g. for DNA K=4
Different rate matrices make different assumptions of substitutions
Jukes Cantor: all substitutions have same rates.
Kimura: transitions (A<->G, C<->T) and transversions (A<->C,A<->T,G<->C,G<->T) have different rates.
Hasegawa, Kishino, Yano (HKY, all substitutions have different rates).

6. Jukes Cantor Rate matrix
Simplest possible rate matrix for DNA sequence evolution
Assumes all bases change at the same rate A T G C A
Mutations are occurring from rows to columns. T G C

7. Conditional probabilities from Jukes Cantor
The conditional probability matrix, P(a|b,t) has a similar form as the rate matrix A T G C A T G C
P(G|C,t)
Equilibrium distribution: ¼ for all bases

8. Searching phylogenetic tree space with maximum likelihood
As in the maximum parsimony case we need to
Score a tree
Search over the space of possible trees
Score a given tree
Branch lengths are parameters
Estimate the branch lengths to maximize the likelihood of data given tree
Search over trees
Start with an initial tree
A greedy approach of adding a branch that maximizes the likelihood
Neighbor Joining
Revisit using nearest neighbor interchange or subtree grafting approaches until convergence

9. Some advantages of probabilistic approaches
Probabilistic models can be naturally extended to more realistic model
Model site specific parameters
Model gaps
A probabilistic framework can be used to evaluate different models of varying complexity (more parameters)
Different evolutionary models
Easily combined with other probabilistic models
Hidden Markov models

10. Modeling site-specific parameters
Recall we had assumed that the probabilities at each is the same
This could be relaxed by introducing additional parameters per site, ru

11. Probabilistic interpretation of Parsimony
Recall P(a|b,t) is the key quantity of interest
Replace P(a|b,t) by P(a|b) and use –log P(a|b) as the score
Applying the weighted parsimony algorithm on this score to get the minimal cost tree will give an approximation to likelihood
The one associated with the most likely assignment of the ancestral states

12. Bootstrap: Assessing reliability of phylogenetic trees
Bootstrap: a computational strategy used to assess confidence in an estimated quantity
E.g. branch length
Tree branching topology
Generate a bunch of trees, {T1,…,TN}, from N random samples of the data
Sample columns/sites with replacement
Reconstruct a tree from sampled columns
One can estimate the confidence of any tree feature based on the proportion of times the feature is seen in a tree in {T1,…,TN}

13. Example of bootstrap
Ziheng Yang and Bruce Rannala, Nature Reviews Genetics 2012

14. Some common phylogenetic tree construction algorithms
PhyML
Maximum likelihood, Nearest neighbor interchange, subtree pruning and regrafting
RAxML (Randomized Axelerated Maximum Likelihood)
Exists in both sequential and parallel versions
Also does subtree pruning and regrafting
PhyLIP (From Felsenstein)
Package for distance-based, parsimony, ML methods
BEAST (Bayesian)
MCMC based sampling
MrBayes (Bayesian)
Visit here for more

15. Comments about phylogenetic tree construction
Which method to pick?
Neighbor joining: fast, constructs right tree if the distances are additive
Parsimony: does not make any assumption of distances
Probabilistic:
More principled, provides a systematic framework to estimate model parameters
Enables us to quantify uncertainty in the model, evaluate different models of evolution
If ML distances are additive NJ can construct the right tree
If branch lengths are ignored, weighted parsimony and maximum likelihood are equivalent
Search space may be large, but
can find the optimal tree efficiently in some cases
heuristic methods can be applied
Difficult to evaluate inferred phylogenies: ground truth not usually known
can look at agreement across different sources of evidence
can look at repeatability across subsamples of the data (bootstrap)
can look at indirect predictions, e.g. conservation of sites in proteins
Methods could be assessed based on a simulation framework based on a probabilistic model of phylogenies
Phylogenies for bacteria, viruses not so straightforward because of lateral transfer of genetic material