# Maximum Likelihood Tree Estimation. Carrie Tribble IB Feb 2018

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1 Maximum Likelihood Tree Estimation Carrie Tribble IB Feb 2018

2 Outline 1. Tree building process under maximum likelihood 2. Key differences between maximum likelihood and parsimony 3. Some fancy extras

3 It [maximum likelihood] involves finding that evolutionary tree which yields the highest probability of evolving the observed data (Felsenstein 1981).

4 Maximum Likelihood Likelihood = P(D M) Probability Data Model

5 Maximum Likelihood Likelihood = P(D M) Probability Alignment Topology (Ψ), Branch lengths (v), Model Parameters (θ)

6 Toss a coin 10 times: H T H H T T H H H H Probability of heads is p Probability of tails is 1-p What is the probability of our data given our model of a two-sided coin? H T H H T T H H H H L = p * (1-p) * p * p *(1-p) * (1-p) * p * p * p * p L = p 7 * (1-p) 3

7 p = seq(from = 0, to = 1, by = 0.01) L = p^7 * (1 p)^3

8 Our new approach 1. Build a model of character evolution 2. Use that model to calculate the likelihood of a particular phylogeny 3. Find the tree/ parameter values with the highest likelihood (maximum likelihood)

9 Our new approach: Step 1 Build a model of character evolution What is a model? Parameters vs. parameter values A T C G A???? T???? C???? G????

10 Our new approach: Step 1 Build a model of character evolution What is a model? Parameters vs. parameter values A T C G A - a a a T a - a a C a a - a G a a a -

11 Our new approach: Step 1 Build a model of character evolution What is a model? Parameters vs. parameter values A T C G A T C G

12 Our new approach: Step 1 Build a model of character evolution What is a model? Parameters vs. parameter values Modified from Huelsenbeck 2017

13 Q = Jukes Cantor (JC69) A G C T A - μ/3 μ/3 μ/3 G μ/3 - μ/3 μ/3 C μ/3 μ/3 - μ/3 T μ/3 μ/3 μ/3 - Kimura (K80) A G C T A - K 1 1 G K C K T 1 1 K - * μ A <-> G: Transition (purine purine) C <-> T: Transition (pyrimadine pyrimadine) Felsenstein (F81) A G C T A - π G π C π T G π A - π C π T C π A π G - π T T π A π G π C - Hasegawa, Kishino, Yano (HKY85) A G C T A - kπ G π C π T * μ G kπ A - π C π T * μ C π A π G - kπ T T π A π G kπ C - Base frequencies vary

14 General Time Reversible (GTR): A G C T A - απ G βπ C γπ T G απ A - δπ C επ T * μ C βπ A δπ G - ηπ T T γπ A επ G ηπ C - Rate parameters: α = A <-> G β = A <-> C γ = A <-> T δ = C <-> G ε = T <-> G η = T <-> C Stationary base frequencies: π A = frequency of A π G = frequency of G π C = frequency of A π T = frequency of T

15 General Time Reversible (GTR): Markov models are memoryless A G C T A - απ G βπ C γπ T G απ A - δπ C επ T * μ C βπ A δπ G - ηπ T T γπ A επ G ηπ C - Rate parameters: α = A <-> G β = A <-> C γ = A <-> T δ = C <-> G ε = T <-> G η = T <-> C Stationary base frequencies: π A = frequency of A π G = frequency of G π C = frequency of A π T = frequency of T

16

17 Our new approach: Step 2 It [maximum likelihood] involves finding that evolutionary tree which yields the highest probability of evolving the observed data (Felsenstein 1981). GAATC GTATC AAATC GAATC GTATC AAATC GAATC GTATC AAATC Let s start with one site in the alignment

18 Our new approach: Step 2 P ij (t) π s0 Felsenstein 1981

19 Our new approach: Step Probability of node for all possible base pairs Felsenstein s pruning algorithm Figured modified from Huelsenbeck 2017

20 Where do we get P ij (t) and π s0? Q = A G C T A -3λ λ λ λ G λ -3λ λ λ C λ λ -3λ λ T λ λ λ -3λ A G C T P(t) = e Qt = A p 0 (t) p 1 (t) p 1 (t) p 1 (t) G p 1 (t) p 0 (t) p 1 (t) p 1 (t) C p 1 (t) p 1 (t) p 0 (t) p 1 (t) with, p 0 (t) = ¼ + ¾e -4λt p 1 (t) = ¼ - ¼e -4λt T p 1 (t) p 1 (t) p 1 (t) p 0 (t)

21 Pick a tree & parameter values Use Felsenstein s Pruning Algorithm & your model of evolution to calculate the likelihood of a tree for a single site in an alignment Assume all sites in alignment are independent & multiply likelihoods of all sites to calculate the likelihood of the tree given the alignment Vary parameter values for that topology to find the maximum likelihood Pick a new tree, repeat! Overview:

22 This process is often untenable computationally Number of rooted trees increases A LOT with increasing number of taxa multiplying small number -> smaller numbers so we use log likelihood

23

24 Main differences: Branch length matter!!!!! Estimation of parameters Model underlies the process

25 Main differences: Branch length matter!!!!! ATTCG ATTCG Time What happened along the branches? ATTCG ATTCG

26 Main differences: Branch length matter!!!!! ATTCG ATTCG Time C -> G What happened along the branches? ATTCG G -> C ATTCG

27 Main differences: Branch length matter!!!!! Estimation of parameters Model underlies the process

28 Main differences: Branch lengths matter!!!!! Estimation of parameters How does this compare to weighting in parsimony?

29 Fancy Things Error? Rate heterogeneity Model testing Additional models (codon, amino acid, etc.)

30 Get idea of error through bootstrapping But what error is this measuring?

31 Rate Heterogeneity: Partitions gene1 gene2 gene3 gene4 gene5 A G C T A -3λ λ λ λ G λ -3λ λ λ C λ λ -3λ λ T λ λ λ -3λ

32 Rate Heterogeneity: Partitions A G C T A G C T A -3λ λ λ λ A -3λ λ λ λ G λ -3λ λ λ G λ -3λ λ λ C λ λ -3λ λ C λ λ -3λ λ T λ λ λ -3λ T λ λ λ -3λ gene1 gene2 gene3 gene4 gene5 A G C T A -3λ λ λ λ G λ -3λ λ λ C λ λ -3λ λ T λ λ λ -3λ A G C T A -3λ λ λ λ G λ -3λ λ λ C λ λ -3λ λ T λ λ λ -3λ A G C T A -3λ λ λ λ G λ -3λ λ λ C λ λ -3λ λ T λ λ λ -3λ Estimate model parameters separately for different genes!

33 Rate Heterogeneity: Partitions gene1 gene2 gene3 gene4 gene5 Estimate model parameters separately for different codon positions!

34 Rate Heterogeneity: gamma (Γ)

35 Model testing Likelihood ratio test for nested models: LR = 2 * (-lnl1 -lnl2) HKY85 -lnl = GTR -lnl = LR = 2( ) = 4.53, df = 4 critical value (P = 0.05) = 9.49 (~chi 2 distribution)

36 Model testing Aikake Information Criterion (AIC) (non nested): Find model with lowest AIC Models need not be nested Can correct for small sample sizes (AICc) BIC is a similar alternative

37 Additional models Amino Acid Codon non-markov What would a model of morphology look like?

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