# Realistic evolutionary models Marjolijn Elsinga & Lars Hemel

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• Slide 1
• Realistic evolutionary models Marjolijn Elsinga & Lars Hemel
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• Realistic evolutionary models Contents Models with different rates at different sites Models which allow gaps Evaluating different models Break Probabilistic interpretation of Parsimony Maximum Likelihood distances
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• Unrealistic assumptions 1 Same rate of evolution at each site in the substitution matrix - In reality: the structure of proteins and the base pairing of RNA result in different rates 2 Ungapped alignments - Discard useful information given by the pattern of deletions and insertions
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• Different rates in matrix Maximum likelihood, sites are independent X j for j = 1n
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• Different rates in matrix (2) Introduce a site-dependent variable r u
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• Different rates in matrix (3) We dont know r u, so we use a prior Yang [1993] suggests a gamma distribution g(r, , ), with mean = 1 and variance = 1/
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• Problem Number of terms grows exponentially with the number of sequences computationally slow Solution: approximation - Replace integral by a discrete sum - Subdivide domain into m intervals - Let r k denote the mean of the gamma distribution in the kth interval
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• Solution Yang [1993] found m = 3.4 gives a good approximation Only m times as much computation as for non-varying sites
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• Evolutionary models with gaps (1) Idea 1: introduce _ as an extra character of the alphabet of K residues and replace the (KxK) matrix with a (K+1) x (K+1) matrix Drawback: no possibility to assign lower cost to a following gap, gaps are now independent
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• Evolutionary models with gaps (2) Idea 2: Allison, Wallace & Yee [1992] introduce delete and insertion states to ensure affine-type gaps Drawback: computationally intractable
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• Evolutionary models with gaps (3) Idea 3: Thorne, Kishino & Felsenstein [1992] use fragment substitution to get a degree of biological plausibility Drawback: usable for only two sequences
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• Finally Find a way to use affine-type gap penalties in a computationally reasonable way Mitchison & Durbin [1995] made a tree HMM which uses a profile HMM architecture, and treats paths through the model as objects that undergo evolutionary change
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• Assumptions needed again We will use a architecture quite simpler than that of the profile HMM of Krogh et al [1994]: it has only match and delete states Match state: M k Delete state: D k k = position in the model
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• Tree HMM with gaps (1) Sequence y is ancestor of sequence x Both sequences are aligned to the model, so both follow a prescribed path through the model
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• Tree HMM with gaps (2) x emits residu x i at M k y emits residu y j at M k Probability of substitution y j x i is P(x i | y j,t)
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• Tree HMM with gaps (3) What if x goes a different path than y x: M k D k+1 (= MD) y: M k M k+1 (= MM) P(MD|MM, t)
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• Tree HMM with gaps (4) x: D k+1 M k+2 (= DM) y: M k+1 M k+2 (= MM) We assume that the choice between DD and DM is controlled by a mutational process that operates independently from y
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• Substitution matrix The probabilities of transitions of the path of x are given by priors: D k+1 M k+2 has probability q DM
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• How it works At position k: q yj P(x i |y j,t) Transition k k+1: q MM P(MD|MM,t) Transition k+1 k+2: q MM q DM
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• An other example
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• Evaluating models: evidence Comparing models is difficult Compare probabilities: P(D|M 1 ) and P(D|M 2 ) by integrating over all parameters of each model Parameters Prior probabilities P( )
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• Comparing two models Natural way to compare M 1 and M 2 is to compute the posterior probability of M 1
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• Parametric Bootstrap Let be the maximum likelihood of the data D for the model M 1 Let be the maximum likelihood of the data D for the model M 2
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• Parametric bootstrap (2) Simulate datasets D i with the values of the parameters of M 1 that gave the maximum likelihood for D If exceed almost all values of i M 2 captured more aspects of the data that M 1 did not mimic, therefore M 1 is rejected
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• Break
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• Probabilistic interpretation of various models Lars Hemel
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• Overview Review of last weeks method Parsimony Assumptions, Properties Probabilistic interpretation of Parsimony Maximum Likelihood distances Example: Neighbour joining More probabilistic interpretations Sankoff & Cedergren Heins affine cost algorithm Conclusion / Questions?
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• Review Parsimony = Finding a tree which can explain the observed sequences with a minimal number of substitutions
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• Parsimony Remember the following assumptions: Sequences are aligned Alignments do not have gaps Each site is treated independently Further more, many families have: Substitution matrix is multiplicative: Reversibility:
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• Parsimony Basic step: counting the minimal number of changes for one site Final number of substitutions is summing over all the sites Weighted parsimony uses different weights for different substitutions
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• Probabilistic interpretation of parsimony Given: A set of substitution probabilities P(b|a) in which we neglect the dependence on length t Calculate substitution costs S(a,b) = -log P(b|a) Felsenstein [1981] showed that by using these substitution costs, the minimal cost at site u for the whole tree T obtained by the weighted parsimony algorithm is regarded as an approximation to the likelihood
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• Probabilistic interpretation of parsimony Testing performance for tree-building algorithms can be done by generating trees probabilistic with sampling and then see how often a given algorithm reconstructs them correctly Sampling is done as follows: Pick a residue a at the root with probability Accept substitution to b along the edge down to node i with probability repetitive Sequences of length N are generated by N independent repetitions of this procedure Maximum likelihood should reconstruct the correct tree for large N
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• Probabilistic interpretation of parsimony Suppose we have tree T, with the following edgelengths 0.09 0.1 0.3 And substitutionmatrix with p=0.3 for leaves 1,3 and p=0.1 for 2 and 4 1 2 4 3
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• Probabilistic interpretation of parsimony Tree with n leaves has (2n-5)!! unrooted trees 1 2 3 4 1 2 4 3 12 3 4
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• Probabilistic interpretation of parsimony Parsimony can constructs the wrong tree even for large N N 20419339242 100638204158 5009046135 200099730 N 20396378224 10040551579 5004045942 20003536460 Parsimony Maximum likelihood
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• Probabilistic interpretation of parsimony Suppose the following example: A tree with A,A,B,B at the places 1,2,3 and 4 A A B B
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• Probabilistic interpretation of parsimony With parsimony the number of substitutions are calculated AA B B A A B B A A A B 2 1 Parsimony constructs the right tree with 1 substitution more often than the left tree with 2
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• Maximum Likelihood distances Suppose tree T, edge lengths and sampled sequences at the leafs Well try to compute the distance between and
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• By multiplicativety Maximum Likelihood distances
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• By reversibility and multiplicativity
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• Maximum Likelihood distances
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• ML distances between leaf sequences are close to additive, given large amount of data
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• Example: Neighbour joining i j k m
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• Use Maximum Likelihood distances Suppose we have a multiplicative reversible model Suppose we have plenty of data The underlying probabilistic model is correct Then Neighbour joining will construct any tree correctly.
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• Example: Neighbour joining Neighbour joining using ML distances It constructs the correct tree where Parsimony failed N 20477301222 100635231134 5008968519 200099750
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• More probabilistic interpretations Sankoff & Cedergren Simultaneously aligning sequences and finding its phylogeny, by using a character substitution model Probabilistic when scores are interpreted as log probabilities and if the procedure is additive in stead of maximizing. Allison, Wallace & Yee [1992] But as the original S&C method it is not practical for most problems.
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• More probabilistic interpretations Heins affine cost algorithm Simultaneously aligning sequences and finding its phylogeny, by using affine gap penalties Probabilistic when scores are interpreted as log probabilities and if the

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