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Bayesian Evolutionary Distance P. Agarwal and D.J. States. Bayesian evolutionary distance. Journal of Computational Biology 3(1):1— 17, 1996

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Determining time of divergence Goal: Determine when two aligned sequences X and Y diverged from a common ancestor AGTTGAC ACTTGCC Model: –Mutation only –Independence –Markov process

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Divergence points have different probabilities Probability X Y Ancestor time

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DNA PAM matrices Similar to Dayhoff PAM matrices PAM 1 corresponds to 1% mutation –1% change ≈ 10 million years Simplification: uniform mutation rates among nucleotides: –m ij = if i = j –m ij = if i j Can modify to handle different transition/transversion rates –Transitions (A G or C T) have higher probability than transversions PAM x = (PAM 1) x

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DNA PAM A CTG T A G A

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DNA PAM x (x) (x) (x) (x) (x) (x) (x) A CTG T A G A

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DNA PAM x As x , (x) and (x) 1/4 Assume p i = ¼ for i ={A,C,T,G} Leads to simple match/mismatch scoring scheme

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DNA PAM x: Scoring

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DNA PAM PAM Dist (x) Match score (bits) Mismatch score (bits)

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DNA PAM n: Scoring Log-odds score of alignment of length n with k mismatches: Odds score of same alignment: )(4log)( )(4 )( xx xx

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Probability of k mismatches at distance x Note: Need odds score here, not log-odds!

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Expected evolutionary distance given k mismatches Over all distances By Bayes’ Thm: Conditional expectation From odds scores ??

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Assumptions Consider only a finite number of values of x; e.g., 1, 10, 25,50, etc. –In theory, could consider any number of values “Flat prior:” All values of x are equally likely –If M values are considered, Pr(x) = 1/M

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Calculating Pr(k) and Pr(x|k)

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Calculating the distance Fraction of the probability of k mismatches that comes from assuming distance is x

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Ungapped local alignments Only matches and mismatches — no gaps An ungapped local alignment of sequences X and Y is a pair of equal-length substrings of X and Y X Y

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Ungapped local alignments P. Agarwal and D.J. States. Bayesian evolutionary distance. Journal of Computational Biology 3(1):1—17, matches 2 mismatches 34 matches 11 mismatches A:A: B:B:

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Which alignment is better? Answer depends on evolutionary distance

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