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Multiple Sequence Alignment

Multiple Sequence Alignment. By Yuan Li. Multiple Sequence Alignment. Lots of foundational problems in molecular biology are NP-hard Multiple Sequence Alignment Phylogeny Construction DNA sequencing (Shorest Common Superstring) ‏ RNA Structure Crossing Alignment K-mean Clustering.

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Multiple Sequence Alignment

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  1. Multiple Sequence Alignment By Yuan Li

  2. Multiple Sequence Alignment • Lots of foundational problems in molecular biology are NP-hard • Multiple Sequence Alignment • Phylogeny Construction • DNA sequencing (Shorest Common Superstring)‏ • RNA Structure Crossing Alignment • K-mean Clustering

  3. Multiple Sequence Alignment • A sequence alignment of three or more biological sequences, generally protein, DNA, or RNA • The input set of sequences share a lineage and a common ancestor • Sequence homology can be inferred and phylogenetic analysis can be conducted to MSA • Be used to access sequence conservation of proteins domain, DNA primary/secondary/tertiary structures

  4. Pairwise Alignment • Mutations: substitution, insertion, deletion • Input: Given two sequences, s1 and s2 • Output: The least number of mutations needed to convert s1 to s2, which is also the distance between s1 and s2 • Example: • S1 = AAGG–TGC • S2 = A– GTATCC • d(s1, s2) = 4

  5. Multiple Sequence Alignment • Input: a set of n sequences, {s1,s2,...,sn} • Output: a n*L matrix, so that a certain criteria is optimal • Input: GTAAC, TAAC, GTAC • Output: • GTAAC • - TAAC • GTA- C • Criteria: sum of pairs score, star align, tree align

  6. Star Align - Optimization • Input: Given a set of strings S={s1, s2,..., sn} • Output: a optimal string c, such that the sum of distance between c and si (where 1<=i<=n), is minimum.

  7. Star Align - Decision • Input: Given a set of strings S={s1, s2,..., sn}, and a interger k • Question: Is there a string c, such that the sum of distance between c and si (where 1<=i<=n), is less or equal to k?

  8. NPC Problem • 1) It is a decision problem • 2) It is in the set NP • Given a string c, the sum of distance between c and every string in S can be calculated in polynomial time and thus verify the correctness • 3) Reduce to Vertex Cover • Given ins(VC), an arbitrary instance of VC, construct an instance of star align, ins(SA)‏ • Proof that ins(VC) is true iff ins(SA) is true

  9. Star Alignment A set of strings, S A optimal string, c=DDCDD Reduction • Vertex Cover • A graph (V,E)‏ • |V|=n, |E|=m • Minimum cover, v'

  10. Construction Idea • Define Three types of Components • Base Component = {E,G} • Selection Component = {E,S(i,j)} • Ground Component = {G} • Construction • vertice--> {E,G} • edge(Vi,Vj)-->{E,S(i,j)}

  11. Definition • Paddings, P • 0s 1s 0s, s>=(n+1)‏ • 0..0 1..1 0..0 • Block1, B1 (vertex position = 1)‏ • P1P, i.e. 0..0 1..1 0..0 1 0..0 1..1 0..0 • Block0, B0 (vertex position = 0)‏ • P0P, i.e. 0..0 1..1 0..0 0 0..0 1..1 0..0 • String for vertex i, Vi • (B0)i-1 B1(B0)n-i

  12. Definition • Delimiter String, D • 1111111...111111, of length |Vi| • Cover String, C • (B1|B0)n • Base String, c = DDCDD • Enforcing String, E = DD (B1)n DD • Ground String, G = DD (B0)n DD • Selection String, S(i,j) = ViDVj

  13. Comparision

  14. Base Component • Base Component {E,G} • # = n, for each vertex, construct a base component {E, G} • E = DD (B1)n DD • G = DD (B0)n DD • Lemma • The only optimal alignment of E and G is the direct match • If d(E,x)+d(G,x)<d(E,G)+1, x is base string, DDCDD.

  15. Selection Component • Selection Component {E, S(i,j)} • # = m, for each edge(vi,vj), construct a selection component E, S(i,j)‏ • E = DD (B1)n DD, votes 1 in all vertex positions. • S(i,j) = Vi D Vj, votes 0 in all except vertex position i or j, so that either vertex i or vertex j is part of the vertex cover • D D | C | D D • Vi D | Vj | • | Vi | D Vj

  16. Ground Component • Ground Component {G} • # = 1, only construct one ground component • G = DD(B0)n DD • c = DDCDD • d(G, c) means align • ....0....0...0...0...0...0... • ....?....?...?...?...?...?... • G will penalze each 1 in vertex positions, so that the sum of d(c, si) is minimum <--> the size of vertex cover v' is minimum.

  17. Component • Base component {E,G} → c = DDCDD • Selection component {E, S(i,j)} →c <--> Vertex Cover • Ground component {G} →minimum cover

  18. Conclusion • Vertex Cover is a NP-Complete Problem • Vertex Cover can be transformed to Star Alignment in polynomial time • So that Star Alignment is also a NP-Complete Problem

  19. Reference • Isaac Elias, Settling the intractability of multiple alignment, in Proc. of the 14th Ann. Int. Symp. on Algorithms and Computation (ISAAC), 2003, p352--363

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