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phylogeny2

# 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!
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
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 π T π G π C π A π T π G π C π A π T π G π C π A π T π G π C Ρਟ Σਿ ΢ਯ ΢ਯ ΢ਯ ΢ਯ Τ੏ Φ੯ Υ੟ Υ੟ Υ੟ Υ੟ •Assume Pr( i ) constant over time Assume Pr( j | i ) = Pr( j ) = π j (conditional independence)
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 π i i = (1 π i ) π i i = 1 π i 2 i e.g. For π i = 1/4, H = 3/4 Note: “m” means “mutation event occurred”.
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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