CSCI2950-C Lecture 4 DNA Sequencing and Fragment Assembly

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1 CSCI2950-C Lecture 4 DNA Sequencing and Fragment Assembly Ben Raphael Sept. 22, l-mer composition Def: Given string s, the Spectrum ( s, l ) is unordered multiset of all possible (n l + 1) l-mers in a string s of length n The order of individual elements in Spectrum ( s, l ) does not matter For s = TATGGTGC all of the following are equivalent representations of Spectrum ( s, 3 ): {TAT, ATG, TGG, GGT, GTG, TGC} {ATG, GGT, GTG, TAT, TGC, TGG} {TGG, TGC, TAT, GTG, GGT, ATG} 1

2 The SBH Problem Goal: Reconstruct a string from its l-mer composition Input: A multiset S, representing all l-mers from an (unknown) string s Output: String s such that Spectrum ( s,l ) = S SBH: An Example DNA Sequencing AGT CCA ATC ATCCAGT TCC CAG S = { ATC, CCA, CAG, TCC, AGT } 2

3 SBH: Hamiltonian Path Approach S = { ATG AGG TGC TCC GTC GGT GCA CAG } Directed graph G = (S,E). e s,t E if suffix of s = prefix of t, where length(suffix/prefix) = l - 1 H ATG AGG TGC TCC GTC GGT GCA CAG ATG C A G G T C C Path visited every VERTEX once: Hamiltonian Path SBH: Hamiltonian Path Approach A more complicated graph: S = { ATG TGG TGC GTG GGC GCA GCG CGT } 3

4 SBH: Hamiltonian Path Approach S = { ATG TGG TGC GTG GGC GCA GCG CGT } Path 1: ATGCGTGGCA Path 2: ATGGCGTGCA SBH: Eulerian Path Approach S = { ATG, TGG, TGC, GTG, GGC, GCA, GCG, CGT } Vertices correspond to ( l 1 ) mers : { AT, TG, GC, GG, GT, CA, CG } Edges correspond to l mers from S GT CG (l-1) debruijn graph of S: Vertices: (l-1)-mers in S Directed edges: consecutive (l-1) mers in S AT TG GC CA GG Find path that visits every EDGE once 4

5 SBH: Eulerian Path Approach S = {ATG, TGG, TGC, GTG, GGC, GCA, GCG, CGT } Two different paths give different sequence reconstructions: GT CG GT CG AT TG GC CA AT TG GC CA GG GG ATGGCGTGCA ATGCGTGGCA Euler Theorem A graph is balanced if for every vertex the number of incoming edges equals to the number of outgoing edges: in(v)=out(v) Theorem: A connected graph is Eulerian if and only if each of its vertices is balanced. 5

6 Euler Theorem: Proof Eulerian balanced for every edge entering v (incoming edge) there exists an edge leaving v (outgoing edge). Therefore in(v)=out(v) Balanced Eulerian??? Algorithm for Constructing an Eulerian Cycle a. Start with an arbitrary vertex v and form an arbitrary cycle with unused edges until a dead end is reached. Since the graph is Eulerian this dead end is necessarily the starting point, i.e., vertex v. 6

7 Algorithm for Constructing an Eulerian Cycle (cont d) b. If cycle from (a) above is not an Eulerian cycle, it must contain a vertex w, which has untraversed edges. Perform step (a) again, using vertex w as the starting point. Once again, we will end up in the starting vertex w. Algorithm for Constructing an Eulerian Cycle (cont d) c. Combine the cycles from (a) and (b) into a single cycle and iterate step (b). 7

8 Euler Theorem: Extension Theorem: A connected graph has an Eulerian path if and only if it contains at most two semi-balanced vertices and all other vertices are balanced. Back to sequence assembly Overlap Graph Each vertex represents a read from the original sequence. Vertices from repeats are connected to many others. Repeat Repeat Repeat Find a path visiting every VERTEX exactly once: Hamiltonian path problem 8

9 Fragment Assembly: Overlap Graph Find a path visiting every VERTEX exactly once in the OVERLAP graph: Hamiltonian path problem NP-complete: algorithms unknown Repeat Graph: Eulerian Approach Repeat Repeat Repeat Suppose each repeat and unique string (non-repeat) represented by directed edge. Edge of multiplicity 3 Genome sequence is path visiting every EDGE exactly once: Eulerian path problem 9

10 Multiple Repeats Repeat1 Repeat2 Repeat1 Repeat2 Can be easily constructed with any number of repeats Approaches to Fragment Assembly (cont d) Find a path visiting every EDGE exactly once in the REPEAT graph: Eulerian path problem Linear time algorithms are known 10

11 Repeat Graph (a) DNA sequence with a triple repeat R; (b) the layout graph; (c) construction of the de Bruijn graph by gluing repeats; (d) de Bruijn graph. Pevzner P. A. et.al. PNAS 2001;98: Building Repeat Graph Problem: Construct the repeat graph from a collection of reads.? Solution: Break the reads into smaller pieces. 11

12 Building Repeat Graph Reads are constructed from an original sequence in lengths that allow biologists a high level of certainty. They are then broken again into k-mers Repeat Graph Vertices correspond to ( k 1 ) mers in each read Edges correspond to k mers in each read Example: S = ATGGCGTGCA Reads = {ATGGC, GGCGTG, GTGCA} 3-mers = { ATG, TGG, TGC, GTG, GGC, GCA, GCG, CGT } GT CG Two Eulerian paths: (visit every EDGE once) AT TG GC CA ATGCGTGGCA ATGGCGTGCA GG 12

13 Repeat Graph Vertices correspond to ( k 1 ) mers in each read Edges correspond to k mers in each read Example: S = ATGGCGTGCA Reads = {ATGGC, GGCGTG, GTGCA} 3-mers = { ATG, TGG, TGC, GTG, GGC, GCA, GCG, CGT } GT CG AT TG GC CA GG Reads in Repeat Graph S = {ATG, TGG, TGC, GTG, GGC, GCA, GCG, CGT } Two different paths give different sequence reconstructions: GT CG AT TG GC CA GG ATGGCGTGCA 13

14 Reads in Repeat Graph Example: S = ATGGCGTGCA Reads = {ATGGC, GGCGTG, GTGCA} 3-mers = { ATG, TGG, TGC, GTG, GGC, GCA, GCG, CGT } GT CG AT TG GC GG ATGCGTGGCA CA Not all Eulerian paths preserve reads! Eulerian Superpath Example: S = ATGGCGTGCA Reads = {ATGGC, GGCGTG, GTGCA} 3-mers = { ATG, TGG, TGC, GTG, GGC, GCA, GCG, CGT } Eulerian superpath: an Eulerian path that contains set of paths (reads) as subpaths. GT CG AT TG GC CA ATGCGTGGCA ATGGCGTGCA GG 14

15 EULER Fragment Assembly Approach Input: Reads s 1,, s N Further subdivide reads into k-mers (k = 20) Build repeat graph on resulting k-mers Each read is path in resulting graph. Solve Eulerian Superpath Problem. Given an Eulerian graph and a collection of paths in this graph, find an Eulerian path in this graph that contains all these paths as subpaths. Simplifying the Repeat Graph Collapse paths Concat(1,2,3,4) Collapse tree-like structures with unique path Unzip edges A B C D A B C C D 15

16 Additional challenges in EULER Approach 1. Errors in reads 2. Reverse-complement of DNA string 3. Using mate-pair information to simplify the repeat graph. 4. Multiplicities of edges generally unknown (Copy number problem). Sequencing Errors If an error exists in one of the reads, the error will be perpetuated among all of the k-mers broken from that read. 16

17 Error Correction from k-mers Consensus first approach to error correction. Let G k = {k-mers in genome G} Let T = {all k-tuples appearing in > M reads} G k T. Error Correction from k-mers A string s is called a T-string if all its l-tuples belong to T. Spectral Alignment Problem. Given a string s and a spectrum T, find the minimum number of mutations in s that transform s into a T-string. Solving Spectral Alignment Problem attempts to eliminate most point mutation errors before reconstructing the original sequence. Not perfect! 17

18 Forward and Reverse Complements 5 3 DNA is copied in 5 3 direction. 3 5 We obtain reads from both strands of DNA. Do not know strand of origin. s = CAGT s = ACTG (reverse complement) Forward and Reverse Complements 5 3 DNA is copied in 5 3 direction. 3 5 We want assembly of either strand, but NOT a mix of both. 18

19 5 Forward and Reverse Complements 3 In Euler assembler, include reverse complement of each read. 3 5 assume that S contains a complement of every read and that the de Bruijn graph can be partitioned into two subgraphs (the canonical one and its reverse complement) Alternative approaches use bidirected graphs. Sources Serafim Batzoglou CS262_2006/ (Sequencing slides) (Euler slides) Euler assembler. Pevzner, Tang, and Waterman (PNAS 2001): on website 19