Phylogenetic Trees Lecture 3

1. Phylogenetic TreesLecture 3 Based on: Durbin et al Chapter 8 .

2. branch internal node leaf Phylogenetic Tree Assumptions • Topology T : bifurcating • Leaves - 1…N • Internal nodes N+1 … 2N-2 • Lengths t = { ti } for each branch • Phylogenetic tree = (Topology, Lengths) = (T, t )

3. Maximum Likelihood Approach Consider the phylogenetic tree to be a stochastic process. AAA Unobserved AAA AGA AAA Observed AGA AAG GGA The probability of transition from character a to character b is given by parameters b|a. 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 b|a ’s and the qa’s.

4. A A A A G A A A G G G A Maximum Likelihood Approach Assume each site evolves independently of the others. Pr(D|Tree, )=iPr(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(D|Tree, )=iPr(D(i)|Tree, )

5. 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

6. 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.

7. A G   C T  -3 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-Cantor assumes equal rate of change:

8. 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 .

9. Leading to the linear differential equation: S’ (t)  S(t)R With the additional condition that in the limit as t goes to infinity: The Jukes-Cantor model (Cont.) For a short time period , we write: By multiplicatively: S(t+ ) = S(t) S()  S(t)(I+R) Hence: [S(t+ ) - S(t)] / S(t) R

10. 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:

11. Kimura 2-parameter model • Allows a different rate for transitions and transversions.

12. Kimura’s 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:

13. Kimura’s K2P model (Cont.) Leading using similar methods to: Where:

14. A C G T Mutation Probabilities Both models satisfy the following properties: • Lack of memory: • Reversibility: • Exist stationary probabilities { Pa } s. t.

15. Probabilistic Approach • Given P,q, the tree topology and branch lengths, we can compute: x5 t4 x4 t2 t3 t1 x1 x2 x3

16. 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.

17. 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.

18. 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 • Termination:Likelihood is

19. 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)

20. T3 T4 T2 Tn T1 Optimal Tree Search • Perform search over possible topologies Parameter space Parametric optimization (EM) Local Maxima

21. 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