Bayesian Evolutionary Distance

1
Bayesian Evolutionary Distance

• P. Agarwal and D.J. States. Bayesian
  evolutionary distance. Journal of Computational
  Biology 3(1)117, 1996

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Determining time of divergence

• Goal Determine when two aligned sequences X and
  Y diverged from a common ancestor AGTTGAC ACTT
  GCC
• Model
• Mutation only
• Independence
• Markov process

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Divergence points have different probabilities
X
Ancestor
Y
Probability
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
• mij ? if i j
• mij ? if i ?? j
• Can modify to handle different transition/transver
  sion rates
• Transitions (A?G or C?T) have higher probability
  than transversions
• PAM x (PAM 1)x

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DNA PAM 1
A
C
T
G
A
G
T
A
6
DNA PAM x
A
C
T
G
A
G
T
A
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DNA PAM x

• As x ? ?, ?(x) and ?(x) ? 1/4
• Assume pi 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
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DNA PAM n Scoring
Log-odds score of alignment of length n with k
mismatches
Odds score of same alignment
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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
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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(xk)
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Calculating the distance
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Ungapped local alignments
An ungapped local alignment of sequences X and Y
is a pair of equal-length substrings of X and Y
Only matches and mismatches no gaps
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Ungapped local alignments
A
B
P. Agarwal and D.J. States. Bayesian
evolutionary distance. Journal of Computational
Biology 3(1)117, 1996
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Which alignment is better?
Answer depends on evolutionary distance