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PPT – Phylogenetic Trees Lecture 3 PowerPoint presentation | free to download - id: d7212-NTM2Z

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Phylogenetic TreesLecture 3

Based on Durbin et al Chapter 8

Phylogenetic Tree Assumptions

- Topology T bifurcating
- Leaves - 1N
- Internal nodes N1 2N-2
- Lengths t ti for each branch
- Phylogenetic tree (Topology, Lengths) (T, t )

Maximum Likelihood Approach

Consider the phylogenetic tree to be a stochastic

process.

AAA

Unobserved

AAA

AGA

AAA

Observed

AGA

GGA

AAG

The probability of transition from character a to

character b is given by parameters ?ba. The

probability of letter a in the root is qa. These

parameters are defined via rates of change per

time unit times the time unit. Given the

complete tree, the probability of data is defined

by the values of the ?ba s and the qas.

Maximum Likelihood Approach

Assume each site evolves independently of the

others.

Pr(DTree, ?)?i Pr(D(i)Tree, ?)

Write down the likelihood of the data (leaves

sequences) given each tree. When the tree is

not given Search for the tree that maximizes

Pr(DTree, ?)?i Pr(D(i)Tree, ?)

Probabilistic Methods

- The phylogenetic tree represents a generative

probabilistic model (like HMMs) for the observed

sequences. - Background probabilities q( a )
- Mutation probabilities P( a b, t )
- Models for evolutionary mutations
- Jukes Cantor
- Kimura 2-parameter model
- Such models are used to derive the probabilities

Jukes Cantor model

- A model for mutation rates

- Mutation occurs at a constant rate
- Each nucleotide is equally likely to mutate into

any other nucleotide with rate a.

The Jukes-Cantor model (1969)

We need to develop a formula for DNA evolution

via Prob(y x, t) where x and y are taken from

A, C, G, T and t is the time length. Jukes-Cant

or assumes equal rate of change

The Jukes-Cantor model (Cont.)

We denote by S(t) the transition probabilities

We assume the matrix is multiplicative in the

sense that S ( t s ) S ( t ) S ( s ) for

any time lengths s or t .

The Jukes-Cantor model (Cont.)

For a short time period ?, we write

By multiplicatively S(t ?) S(t) S(?) ?

S(t)(IR?)

Hence S(t ?) - S(t) /? ? S(t) R

The Jukes-Cantor model (Cont.)

Substituting S(t) into the differential equation

yields

Yielding the unique solution which is known as

the Jukes-Cantor model

Kimura 2-parameter model

- Allows a different rate for transitions and

transversions.

Kimuras K2P model (1980)

Jukes-Cantor model does not take into account

that transitions rates (between purines) A?G and

(between pyrmidine) C?T are different from

transversions rates of A?C, A?T, C?G,

G?T. Kimura used a different rate matrix

Kimuras K2P model (Cont.)

Leading using similar methods to

Where

Mutation Probabilities

- Both models satisfy the following properties
- Lack of memory
- Reversibility
- Exist stationary probabilities Pa s. t.

Probabilistic Approach

- Given P,q, the tree topology and branch lengths,

we can compute

x5

t4

x4

t2

t3

t1

x1

x2

x3

1. Calculate likelihood for each site on a

specific tree. 2. Sum up the L values for all

sites on the tree. 3. Compare the L value for

all possible trees. 4. Choose tree with highest

L value.

Computing the Tree Likelihood

- We are interested in the probability of observed

data given tree and branch lengths - Computed by summing over internal nodes
- This can be done efficiently using a tree upward

traversal pass.

Tree Likelihood Computation

- Define P( Lk a ) prob. of leaves below node

k given that xk a - Init for leaves P( Lk a ) 1 if xk a 0

otherwise - Iteration if k is node with children i and j

, then - TerminationLikelihood is

Maximum Likelihood (ML)

- Score each tree by
- Assumption of independent positions m
- Branch lengths t can be optimized
- Gradient Ascent
- EM
- We look for the highest scoring tree
- Exhaustive
- Sampling methods (Metropolis)

Optimal Tree Search

- Perform search over possible topologies

Parameter space

Parametric optimization (EM)

Local Maxima

Computational Problem

- Such procedures are computationally expensive!
- Computation of optimal parameters, per candidate,

requires non-trivial optimization step. - Spend non-negligible computation on a candidate,

even if it is a low scoring one. - In practice, such learning procedures can only

consider small sets of candidate structures