phylogeny2 - Distance Metrics Cartesian Distance D( x, y )...

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Distance Metrics Cartesian Distance Manhattan Distance Triangle inequality: D ab +D bc D ac Additive distances Clock-like : D(x,A)=D(y,A) for all x,y descended from common ancestor A, “ultrametric” D ( x , y ) = | x i y i | 2 i D ( x , y ) = | x i y i | i D ( x , y ) = d ij x y
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If multiple mutations occur at one site, we only see one… For very low mutation rates (highly similar sequences), the observed number of mutations is a good estimate of the true number that occurred during evolution, because multiple mutations at one site are unlikely. But for high mutation rates: underestimated!
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Observable Variables for Comparing Two Sequences D : number of observed mutations s : number of sites f i = Pr( i ): fraction of observed nucleotide i
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Hidden Variables: Single Event vs. Time m ij = Pr( j | i ): rate matrix for 1 mutation event π i = Pr( i ): stationary nucleotide distribution H = Pr( i j | mut ): probability of base change λ = Pr( mut | t =1): mutation prob. in time t =1 q ij ( t ) = Pr( j | i,t ): probability of i j in time t F ij ( t ) = Pr( i,j | t ): joint prob. of i : j in time t K : expected number of base changes
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Mutation Rate Matrix: m ij A T G C m TA m AA •Simple Markov model representing a single mutation event: m ij = Pr( j | i ) •Note existence of self-edges m ii , as usual in Markov model. •(If this seems confusing, just think of it as the probability that a mutation is repaired by DNA repair machinery no change )
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Simple Stationary Distribution Assumption A T G C π A π A m = π A T G C A T G C A T G C A T G C Ρ Σ ΢ ΢ ΢ ΢ Τ Φ Υ Υ Υ Υ •Assume Pr( i ) constant over time Assume Pr( j | i ) = Pr( j ) = π j (conditional independence)
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Time Reversibility for a Stationary Distribution • For a stationary distribution, we assume equilibrium: forward process and reverse process are indistinguishable. MRCA MRCA t t 2t
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Probability that a Mutation Event Changes H = Pr( i j | m ) = Pr( j | i , m ) j i Pr( i ) i = π j j i i i = (1 −π i ) i i = 1 i 2 i e.g. For π i = 1/4, H = 3/4 Note: “m” means “mutation event occurred”.
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Poisson Model of Mutation Events per Assuming a Poisson model for the number of mutation events k in a time interval t , with on average λ t events per time t : Pr( k ) = e −λ t ( λ t ) k k !
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This note was uploaded on 04/12/2010 for the course CHEM CHEM 260A taught by Professor Chrislee during the Spring '10 term at UCLA.

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phylogeny2 - Distance Metrics Cartesian Distance D( x, y )...

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