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Sensitivity Analysis for Ungapped Markov Models of Evolution

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  1. Sensitivity Analysis for Ungapped Markov Models of Evolution David Fernández-Baca Department of Computer Science Iowa State University (Joint work with Balaji Venkatachalam) CPM '05

  2. 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 CPM '05

  3. X Y Ungapped local alignments An ungapped local alignmentof sequences X and Y is a pair of equal-length substrings of X and Y Only matches and mismatches — no gaps CPM '05

  4. 23 matches 2 mismatches 34 matches 11 mismatches Ungapped local alignments A: B: P. Agarwal and D.J. States. Bayesian evolutionary distance. Journal of Computational Biology 3(1):1—17, 1996 CPM '05

  5. > 0 < 0 score(B) score(A) / -11/9 Which alignment is better? Score =  ∙ #matches +  ∙ #mismatches In practice, scoring schemes depend on evolutionary distance CPM '05

  6. Log-odds scoring Let qX =base frequency of nucleotide XmXY(t) =Prob(XY mutation in t time units) A be an alignment X1X2X3 XnY1Y2Y3 Yn Then, Log odds score ofA = CPM '05

  7. 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) ∙ #mismatcheswhere(t) = 4 + log r(t) (t) = 4 + log s(t) CPM '05

  8. Scores depend nonlinearly on evolutionary distance CPM '05

  9. This talk • An efficient algorithm to compute optimum alignments for all evolutionary distances • Techniques • Linearization • Geometry • Divide-and-conquer CPM '05

  10. Related Work • Combinatorial/linear scoring schemes: • Waterman, Eggert, and Lander 1992: Problem definition • Gusfield, Balasubramanian, and Naor1994: 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. Sturmfels2004a & b: connections between linear scoring and Markov models CPM '05

  11. 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) CPM '05

  12. Jukes-Cantor transition probability matrix where CPM '05

  13. t = +∞   t = 0  versus  (t) = 4 + log r(t) (t) = 4 + log s(t) CPM '05

  14. Recall: Score(A) =  ∙ #matches +  ∙ #mismatches 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 CPM '05

  15. Theorem Letn 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).  CPM '05

  16. 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 / Re-introducing distance The  vs.  curve intersects every boundary line with slope  (-∞, +1] CPM '05

  17. An algorithm • Start with a simple, but highly parallel, algorithm for fixed-parameter problem • 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 • Construct solution to original problem by finding the optimality regions intersected by the (t), (t) curve CPM '05

  18. A naïve dynamic programming algorithm Y • Let C be the matrix whereCij = score of opt alignment ending at Xi and Yj • Subdiagonals correspond to alignments • Diagonals are independent of each other • Process each diagonal separately • Pick best answer over all diagonals • Total time: O(nm) aattcaattcaatc . . . caatttgtcacttttt . . . X C CPM '05

  19. X(1) X(2) X(1) X(2) Y(1) Y(2) Y(1) Y(2) X(1) X(2) Y(1) Y(2) length of diagonal Divide and conquer for diagonals Split diagonal in half, solve each side recursively, and combine answers. E.g.: X Y T(N) = 2 T(N/2) + O(1)  T(N) = O(N) #subproblems CPM '05

  20. 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 CPM '05

  21. f + g f g Adding piecewise linear functions Time = O(total number of segments) CPM '05

  22. f max (f,g) g Computing the maximum of piecewise linear functions Time = O(total number of segments) CPM '05

  23. #(optimum solutions for diagonal) Analysis • Processing a diagonal: • T(n) = 2 T(n/2) + O(n2/3)T(n) = O(n) • Merging score functions for diagonals: • O(n2/3) line segments per function, m+n-1 diagonals • Total time: O(mn + mn2/3 lg m) CPM '05

  24. ACT AAT AGC Further Results (1): Parametric ancestral reconstruction • Given a phylogeny, find most likely ancestors AAC • Sensitive to edge lengths AAT • Result: O(n) algorithm for uniform model (all edge lengths equal) CPM '05

  25. Further Results (2) • Bounds on number of regions for gapped alignment (indels are allowed) • Lead to algorithms, but not as efficient as ungapped case CPM '05

  26. 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 CPM '05

  27. Thanks to • National Science Foundation • CCR-9988348 • EF-0334832 CPM '05