. Phylogenetic Trees Lecture 13 This class consists of parts of Prof Joe Felsenstein’s lectures 4 and 5 taken from: and on Chapter 8.2 of Durbin et al. Edited by Dan Geiger. Background reading: Durbin et al Chapter 8. NOTE: THE PDF FORMAT INCLUDES MORE SLIDES

2 Three Methods of Tree Construction u Distance- A tree that recursively combines two nodes of the smallest distance. u Parsimony – A tree with a total minimum number of character changes between nodes. u Maximum likelihood - Finding the best Bayesian network of a tree shape. The method of choice nowadays. Most known and useful software called phylip uses this method.

3 Maximum Likelihood Approach Consider the phylogenetic tree to be a stochastic process. AGA GGA AAA AAG AAA AGA AAA 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 q a (written  a in Felsenstein’s slides ). 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 q a ’s. Observed Unobserved

4 Maximum Likelihood Approach Assume each site evolves independently of the others. A G A A Write down the likelihood of the data (leaves sequences) given each tree. Use EM to estimate the  b|a parameters. When the tree is not given: Search for the tree that maximizes Pr(D|Tree,  EM )=  i Pr(D (i) |Tree,  EM ) G G A A A A A G Pr(D|Tree,  )=  i Pr(D (i) |Tree,  )

5 The Jukes-Cantor model (1969) We need to develop a formula for DNA evolution via Pr(y|x,t) where x and y are taken from {A,C,G,T} and t is the time length. Jukes-Cantor assume equal rate of change: GA TC    -3 

6 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.

7 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 Leading to the linear differential equation: S`(t)  S(t)R With the additional condition that in the limit as t goes to infinity:

8 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:

9 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:

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

11 Hasegawa, Kishino & Yano model (1985) Still the equilibrium probabilities are all ¼ in Kimura’s model, despite the facts that in many organisms show strong bias in their AT to CG ratio. HKY’s model takes care of this. Also Felsenstein’s model F84 takes care of this problem. There are other models as well, the most general of which is a matrix where all rates of change are distinct (12 parameters). The following chart shows relationships among most used models.