Parametric and Inverse Parametric Sequence Alignment

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1 Parametric and Inverse Parametric Sequence Alignment Nan Lu, School of Mathematics, Georgia Institute of Technology School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 1

2 Parametric optimization of sequence alignment In this section, we will describe the decomposition of the parametric space for two-parameter global alignment. The objective function, therefore, is max α 0,β 0 (w αx βy), where w, x, y are the numbers of matches, mismatches,indels, respectively. We are interested in bounding the number of regions in the α, β plane. And the final conclusion can be obtained by the following lemmas. Lemma 1 For any alignment A with corresponding tuple (w, x, y), 2w + 2x + y = N, where N = n + m is the sum of sequence lengths. Without loss of generality, we may assume m n. Lemma 2 For any alignment A, w + x n. Lemma 3 For any alignment A, m n y m + n. School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 2

3 Lemma 4 In all alignments of two sequences, y is always odd or even depending on whether m + n is odd or even. There are therefore only n + 1 different values for y. The proof of this lemma follows from Lemma 1 and Lemma 3. Theorem 5 Any line forming a boundary between two regions is of the form β = c + (c )α, for some c > 1 2. Proof Since the value of an alignment is always a linear function of the parameters, it follows that the optimal regions, which are bounded by the intersection of hyperplanes, are all convex polygons. When α = 1, β = 1 2, any alignment has value w + x + y 2 = n+m 2, which implies all boundary lines pass through (α, β) = ( 1, 1 2 ). School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 3

4 Let β = c + (c )α for some c, if c 1 2, then β is negative whenever α is nonnegative. Lemma 6 Along any horizontal line we never encounter breakpoints in the region α > 2β. This is because the optimal alignment in this region will contain no mismatch. Lemma 7 There are at most n + 1 regions. Proof Lemma 6 with β = 0 shows that there is no break points along positive α axis. So all the region boundaries intersect with positive β axis. The value of any alignment along positive β axis is w βy, and y only has n + 1 values. Theorem 8 The number of regions is bounded by O(n 2/3 ). School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 4

5 Proof We define ratio form of the boundary line, namely, β = w i w j y i y j + x j x i y i y j α. Let x i = x i+1 x i and y i = y i y i+1, so we can identity the slope of a boundary line with a pair of ( x i, y i ). Moreover, we can make each y i to be positive. Since i x i n and i y i (m + n) (m n) = 2n, 3n k i=1 ( x i + y i ) 0. For all t, there are at most t 1 distinct pairs of ( x i, y i ), such that x i + y i = t. School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 5

6 Let s be the largest value of t, then we have 3n s t=1 t(t 1) = 1 (s 1)s(s + 1) 3 which implies s = O(n 1 3). k s+1 (t 1) = t=1 s(s + 1) 2 = O(s 2 ) = O(n 2 3 ) Theorem 9 All regions in the decomposition can be simply found in O(nm) time. Therefore, the entire decomposition can be found in O(n 5 3m) time. School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 6

7 Multiple sequence alignment using partial order graphs Figure 1: Construction of the partial order bifurcating graph In this paper, the authors use the standard dynamic programming to compute the value of each alignment and partial order graphs to keep all the information of pair-wise alignment. School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 7

An Eulerian path approach to local multiple alignment for DNA sequences Figure 2: Three DNA sequences are built into a three-tuple De Bruijn graph by gluing identical edges and vertices.

8 An Eulerian path approach to local multiple alignment for DNA sequences Figure 2: Three DNA sequences are built into a three-tuple De Bruijn graph by gluing identical edges and vertices. Each sequence is a path traversing the graph. The picture shows the procedure of constructing a De Bruijn Graph. School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 8

9 References [1] Gusfield, D.: Balasubramanian, K.; Naor, D., Parametric optimization of sequence alignment. Algorithmica 12 (1994), no. 4-5, [2] Waterman, Michael; Eggert, Mark; Lander, Eric, Parametric sequence comparisons, Proc. Natl. Acad. Sci. USA 89(1992) [3] Lee, Christopher; Grasso, Catherine; Sharlow, Mark, Multiple sequence alignment using partial order graphs, Bioinformatics 2002 Mar;18(3): [4] Zhang, Yu; Waterman, Michael, An Eulerian path approach to local multiple alignment for DNA sequences, Proc Natl Acad Sci USA 2005 Feb 1;102(5): [5] Sun, Fangting; Fernndez-Baca, David; Yu, Wei Inverse parametric sequence alignment. J. Algorithms 53 (2004), no. 1, School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 9

10 [6] Pevzner, P., Tang, H., and Waterman, M.S. (2001), An Eulerian path approach to DNA fragment assembly, Proc. Natl. Acad. Sci. USA [7] W.M. Fitch and T.F. Smith, Optimal Sequence Alignments, Proc. Natl. Acad. Sci USA 80(1983) [8] Pachter L, Sturmfels B. Parametric inference for biological sequence analysis, textitproc Natl Acad Sci USA 2004 Nov 16;101(46): [9] Lee, Christopher, Generating consensus sequences from partial order multiple sequence alignment graphs, Bioinformatics 2003 May 22;19(8): [10] Grasso C, Lee C., Combining partial order alignment and progressive multiple sequence alignment increases alignment speed and scalability to very large alignment problems, Bioinformatics 2004 Jul 10;20(10): [11] Fernández-Baca, David; Seppäläinen, Timo; Slutzki, Giora Parametric School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 10

11 multiple sequence alignment and phylogeny construction. J. Discrete Algorithms 2 (2004), no. 2 [12] Yap, Roland H.C., Parametric sequence alignment with constraints, Bioinformatics Constraints 6 (2001), no. 2-3, [13] Waterman, Michael S.; Perlwitz, Marcela D. Line geometries for sequence comparisons. Bull. Math. Biol. 46 (1984), no. 4, [14] Waterman, Michael S. General methods of sequence comparison. Bull. Math. Biol. 46 (1984), no. 4, [15] Waterman, Michael S. Efficient sequence alignment algorithms. J. Theoret. Biol. 108 (1984), no. 3, School of Mathematics, Georgia Institute of Technology 22:36, April 9, 2008 First Previous Next Last 11

Lecture 12 March 4th

Math 239: Discrete Mathematics for the Life Sciences Spring 2008 Lecture 12 March 4th Lecturer: Lior Pachter Scribe/ Editor: Wenjing Zheng/ Shaowei Lin 12.1 Alignment Polytopes Recall that the alignment