Sensitivity Analysis for
Ungapped Markov Models of
Evolution
David Fernández-Baca
Department of Computer Science
Iowa State University
(Joint work with Balaji Venkatachalam)
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Motivation
• Alignment scoring schemes are often
based on Markov models of evolution
• Optimum alignment depends on
evolutionary distance
• Our goal: Understand how optimum
alignments are affected by choice of
evolutionary 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
X
Y
Only matches and mismatches — no gaps
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Ungapped local alignments
23 matches
2 mismatches
A:
34 matches
11 mismatches
B:
P. Agarwal and D.J. States. Bayesian evolutionary distance.
Journal of Computational Biology 3(1):1—17, 1996
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Which alignment is better?
Score =  ∙ #matches +  ∙ #mismatches
>0
<0
score(B)
score(A)
/
-11/9
In practice, scoring schemes depend on
evolutionary distance
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Log-odds scoring
Let
qX = base frequency of nucleotide X
mXY (t) = Prob(XY mutation in t time units)
A be an alignment
X1X2X3 Xn
Y1Y2Y3 Yn
Then,
mXiYi (t )
Log odds score of A = log 
i
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qXi  qYi
Log-odds scoring
• Simplest model:
– mXX(t) = r(t) for all X
– mXY(t) = s(t) for all X  Y
– qX = ¼ for all X
• Log-odds score of alignment:
(t) ∙ #matches + (t) ∙ #mismatches
where
(t) = 4 + log r(t)
(t) = 4 + log s(t)
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Scores depend nonlinearly on
evolutionary distance
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This talk
• An efficient algorithm to compute
optimum alignments for all evolutionary
distances
• Techniques
– Linearization
– Geometry
– Divide-and-conquer
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Related Work
• Combinatorial/linear scoring schemes:
– Waterman, Eggert, and Lander 1992: Problem
definition
– Gusfield, Balasubramanian, and Naor 1994: Bounds
on number of optimality regions for pairwise
alignment
– F-B, Seppäläinen, and Slutzki 2004: Generalization
to multiple and phylogenetic alignment
• Sensitivity analysis for statistical models:
– P. Agarwal and D.J. States 1996
– L. Pachter and B. Sturmfels 2004a & b:
connections between linear scoring and Markov
models
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A simple Markov model of
evolution
• Sites evolve independently through
mutation according to a Markov process
• For each site:
– Transition probability matrix:
M = [mij],
i, j {A, C, T, G}
where
mij = Prob(i  j mutation in 1 time unit)
– Transition matrix for t time units is M(t)
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Jukes-Cantor transition
probability matrix
 r(t)

( t)  s(t)
M 
s(t)

 s(t)
where
s(t) s(t) s(t) 

r(t) s(t) s(t) 
s(t) r(t) s(t) 

s(t) s(t) r(t) 

1
r (t )  1  3e  4t
4
1
s (t )  1  e  4t
4

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

 versus 
t = +∞
(t) = 4 + log r(t)
(t) = 4 + log s(t)


t=0
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Linearization
• Allow  and  to vary arbitrarily,
ignoring that they
– are functions of t and
– must satisfy laws of probability
• Result is a linear parametric problem
Recall:
Score(A) =  ∙ #matches +  ∙ #mismatches
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Theorem
Let n be the length of
the shorter sequence.
Then,
(ii) The parameter
space decomposition
looks like this:
(i) The number of
distinct optimal
solutions over all
values of  and  is
O(n2/3).
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

Re-introducing distance
The  vs.  curve
intersects every boundary
line with slope  (-∞, +1]
The optimum solutions for
t = 0 to + are found by
varying  / from - to 1
Non-linear problem in t reduces to a linear
one-parameter problem in  /
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An algorithm
1. Start with a simple, but highly parallel,
algorithm for fixed-parameter problem
2. Lift the fixed-parameter algorithm
•
•
Lifted algorithm runs simultaneously for all
parameter values in linearized problem
Output: A decomposition of parameter space into
optimality regions
3. Construct solution to original problem by
finding the optimality regions intersected by
the (t), (t) curve
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A naïve dynamic programming
algorithm
aattcaattcaatc . . .
X
– Process each diagonal
separately
– Pick best answer over all
diagonals
• Total time: O(nm)
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caatttgtcacttttt . . .
• Let C be the matrix where
Cij = score of opt alignment
ending at Xi and Yj
• Subdiagonals correspond to
alignments
• Diagonals are independent of
each other
Y
C
Divide and conquer for diagonals
Split diagonal in half, solve each side recursively,
and combine answers. E.g.:
X
Y
X(1)
Y(1)
X(2)
Y(2)
X(1)
Y(1)
X(1)
Y(1)
X(2)
Y(2)
T(N) = 2 T(N/2) + O(1)
length of diagonal
X(2)
Y(2)
 T(N) = O(N)
#subproblems
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Lifting
• Run naïve DP algorithm for all
parameter values by manipulating
piecewise linear functions instead of
numbers:
– “+”  “+” for piecewise linear functions
– “max”  “max” of piecewise linear
functions
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Adding piecewise linear functions
f+g
f
g
Time = O(total number of segments)
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Computing the maximum of
piecewise linear functions
max (f,g)
f
g
Time = O(total number of segments)
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Analysis
• Processing a diagonal:
– T(n) = 2 T(n/2) + O(n2/3)
 T(n) = O(n)
#(optimum solutions
for diagonal)
• Merging score functions for diagonals:
– O(n2/3) line segments per function, m+n-1
diagonals
– Total time: O(mn + mn2/3 lg m)
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Further Results (1): Parametric
ancestral reconstruction
• Given a phylogeny,
find most likely
ancestors
• Sensitive to edge
lengths
AAC
AAT
• Result: O(n) algorithm
for uniform model (all
edge lengths equal)
ACT
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AAT
AGC
Further Results (2)
• Bounds on number of regions for gapped
alignment (indels are allowed)
– Lead to algorithms, but not as efficient as
ungapped case
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Open Problems
• Tight bounds on size of parameter
space decomposition
• Evolutionary trees with different
branch lengths
• Efficient sensitivity analysis for gapped
models
• Evaluation of sensitivity to changes in
structure and parameters
– Useful in branch-swapping
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Thanks to
• National Science Foundation
– CCR-9988348
– EF-0334832
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