How should we go about modeling this? Model parameters? Time Substitution rate Can we observe time or subst. rate? What can we observe?

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1 How should we go about modeling this? gorilla GAAGTCCTTGAGAAATAAACTGCACACACTGG orangutan GGACTCCTTGAGAAATAAACTGCACACACTGG Model parameters? Time Substitution rate Can we observe time or subst. rate? What can we observe? 1

2 Maximum likelihood estimation First 32 nucleotides of the ψη-globin gene of gorilla and orangutan: gorilla GAAGTCCTTGAGAAATAAACTGCACACACTGG orangutan GGACTCCTTGAGAAATAAACTGCACACACTGG logl maximum likelihood estimate (MLE) of v is v plot of log-likelihood as a function of branch length v Knowing that v = rate time, Why does the curve drop as you move left of the MLE? Why does the curve drop as you move right of the MLE? 2

3 What does this mean? Why is this number negative? From A C G T JC69 rate matrix 3 3 To A C G T 3 1 parameter: α 3 Jukes, T. H., and C. R. Cantor Evolution of protein molecules. Pages in H. N. Munro (ed.), Mammalian Protein Metabolism. Academic Press, New York. 3

4 Equilibrium Frequencies A sequence consisting only of A... C T G A Perfume bottle broken, and perfume quickly fills the room 4

5 Equilibrium Frequencies If all doors are suddenly opened, perfume will spread by diffusion to the other rooms... C T G A The instant the doors open, the rate away from A is 3α (i.e. rate = -3α) 5

6 Equilibrium Frequencies C T G A Sequence now contains a few Cs, Gs, and Ts... AATAAAAAAAAAAAA AAAACAAAAAATAAA AAAAAAACGAAGAAA AAATAAAAAAAAAAA As perfume spreads by diffusion, the difference in concentration among rooms decreases... 6

7 Equilibrium Frequencies C T G A Sequence contains a mixture of about equal quantities A, C, G and T CAGAATCGAGCAGCT TGACTACGTCATGTG GTTGCGCCGCAACGC CATATACCGCCGACT AGTTTGAGGGCGGTT AGGGCTCGGTTCGTA CATCGTATAAACATT After a long time, equilibrium (=stationarity) is achieved. 7

8 Stationarity Assumed Models assume stationarity, which means that the state frequencies do not change across the tree Models assume that substitution process has reached equilibrium by this point 8

9 From 2 parameters: α β To A C G T K80 (or K2P) rate matrix A C G T transition rate transversion rate Kimura, M A simple method for estimating evolutionary rate of base substitutions through comparative studies of nucleotide sequences. Journal of Molecular Evolution 16:

10 K80 rate matrix (looks different, but actually the same) 2 parameters: κ β A C G T A C G T ( + 2) ( + 2) ( + 2) ( + 2) All I ve done is define κ to be the transition/transversion rate ratio = Note: the K80 model is identical to the JC69 model if κ = 1 (α = β) 10

11 Likelihood Surface when K80 true K80 model (entire 2d space) Based on simulated data: sequence length = 500 sites true branch length = 0.15 true kappa = 5.0 JC69 model (just this 1d line) apple (trs/trv rate ratio) (branch length) 11

12 Likelihood Surface when JC true Based on simulated data: sequence length = 500 sites true branch length = 0.15 true kappa = 1.0 K80 model (entire 2d space) JC69 model (just this 1d line) apple (trs/trv rate ratio) (branch length) 12

13 F81 rate matrix 4 parameters: µ πa πc πg A C G T A C G T µ(1 A ) C µ G µ T µ A µ µ(1 C ) G µ T µ A µ C µ µ(1 G ) T µ A µ C µ G µ µ(1 T ) Note: the F81 model is identical to the JC69 model if all base frequencies are equal Felsenstein, J Evolutionary trees from DNA sequences: a maximum likelihood approach. Journal of Molecular Evolution 17:

14 A C G T HKY85 rate matrix A C G T C G T A G T A C T A C G 5 parameters: κ β πa πc πg A dash means equal to negative sum of other elements on the same row Note: the HKY85 model is identical to the F81 model if κ = 1. If, in addition, all base frequencies are equal, it is identical to JC69. Hasegawa, M., H. Kishino, and T. Yano Dating of the human-ape splitting by a molecular clock of mitochondrial DNA. Journal of Molecular Evolution 21:

15 A C G T GTR rate matrix A C G T C aµ G bµ T cµ A aµ G dµ T eµ A bµ C dµ T fµ A cµ C eµ G fµ 9 parameters: π A π C π G a b c d e µ Identical to the F81 model if a = b = c = d = e = f = 1. If, in addition, all the base frequencies are equal, GTR is identical to JC69. If a = c = d = f = β and b = e = κβ, GTR becomes the HKY85 model. Lanave, C., G. Preparata, C. Saccone, and G. Serio A new method for calculating evolutionary substitution rates. Journal of Molecular Evolution 20:

16 So, how do you turn likelihoods into probabilities? this is the likelihood surface: Pr(data κ,ν) want probability surface: Pr(κ,ν data) apple 16

17 Kinds of probabilities B = Black S = Solid W = White D = Dotted Marginal probabilities: Joint probabilities: 17

18 Kinds of probabilities (continued) Conditional probability 18

19 Bayes rule provides a way to calculate conditional probabilities 19

20 Bayes rule shows how to turn Pr(D B) into Pr(B D) 20

21 The marginal probability of D is the sum of all joint probabilities involving D 21

22 Bayes rule in statistics Likelihood of hypothesis θ Prior probability of hypothesis θ Pr( D) = Pr(D ) Pr( ) Pr(D ) Pr( ) Posterior probability of hypothesis θ Marginal probability of the data (marginalizing over hypotheses) 22

23 Moving through treespace The Larget-Simon move X Step 1: Pick 3 contiguous edges randomly, defining two subtrees, X and Y Y *Larget, B., and D. L. Simon Markov chain monte carlo algorithms for the Bayesian analysis of phylogenetic trees. Molecular Biology and Evolution 16: See also: Holder et al Syst. Biol. 54:

24 Moving through treespace The Larget-Simon move YY X X Step 1: Pick 3 contiguous edges randomly, defining two subtrees, X and Y Step 2: Shrink or grow selected 3-edge segment by a random amount 24

25 Moving through treespace The Larget-Simon move Y X Step 1: Pick 3 contiguous edges randomly, defining two subtrees, X and Y Step 2: Shrink or grow selected 3-edge segment by a random amount 25

26 Moving through treespace The Larget-Simon move Y Step 1: Pick 3 contiguous edges randomly, defining two subtrees, X and Y Step 2: Shrink or grow selected 3-edge segment by a random amount Step 3: Choose X or Y randomly, then reposition randomly 26

27 Moving through treespace The Larget-Simon move Proposed new tree: 3 edge lengths have changed and the topology differs by one NNI rearrangement Step 1: Pick 3 contiguous edges randomly, defining two subtrees, X and Y Step 2: Shrink or grow selected 3-edge segment by a random amount Step 3: Choose X or Y randomly, then reposition randomly 27

28 Moving through treespace Current tree Proposed tree log-posterior = log-posterior = (better, so accept) 28