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 1p What is the probability of our data given our model of a twosided coin? H T H H T T H H H H L = p * (1p) * p * p *(1p) * (1p) * p * p * p * p L = p 7 * (1p) 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 nonmarkov What would a model of morphology look like?
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