Graph Algorithms in Bioinformatics

Transcription

1 Graph Algorithms in Bioinformatics Bioinformatics: Issues and Algorithms CSE Fall 2007 Lecture 13 Lopresti Fall 2007 Lecture

2 Outline Introduction to graph theory Eulerian & Hamiltonian Cycle Problems Benzer experiment and interval graphs DNA sequencing The Shortest Superstring & Traveling Salesman Problems Sequencing by hybridization Lopresti Fall 2007 Lecture

3 The Bridge Obsession Problem Find a tour crossing every bridge just once. Leonhard Euler, 1735 Bridges of Königsberg Lopresti Fall 2007 Lecture

4 Eulerian Cycle Problem Find a cycle that visits every edge exactly once. Algorithm for solution has linear time complexity. More complicated Königsberg Lopresti Fall 2007 Lecture

5 Mapping problems to graphs Arthur Cayley studied chemical structures of hydrocarbons in mid-1800's. He used trees (acyclic connected graphs) to enumerate structural isomers. Lopresti Fall 2007 Lecture

6 Hamiltonian Cycle Problem Find a cycle that visits every vertex exactly once. NP-complete (i.e., a very hard problem to solve). Game invented by Sir William Hamilton in 1857 Lopresti Fall 2007 Lecture

7 Beginning of graph theory in biology Benzer s work: Developed deletion mapping. Proved linearity of gene. Demonstrated internal structure of gene. Seymour Benzer (1950's) Lopresti Fall 2007 Lecture

8 Viruses attack bacteria Normally bacteriophage T4 kills bacteria. However if T4 is mutated (e.g., an important gene is deleted), it gets disabled and loses ability to kill bacteria. Suppose bacteria is infected with two different mutants, each of which is disabled. Would the bacteria still survive? Amazingly, a pair of disable viruses can kill a bacteria even if each of them is disabled. How can this be explained? Lopresti Fall 2007 Lecture

9 Benzer s experiment Idea: infect bacteria with pairs of mutant T4 bacteriophage (virus). Each T4 mutant has an unknown interval deleted from its genome. If two intervals overlap: T4 pair is missing part of its genome and is disabled bacteria survive. If two intervals do not overlap: T4 pair has its entire genome and is enabled bacteria die. Lopresti Fall 2007 Lecture

10 Benzer s experiment Complementation between pairs of mutant T4 bacteriophages Lopresti Fall 2007 Lecture

11 Benzer s experiment and interval graphs Construct interval graph: each T4 mutant is a vertex; place edge between mutant pairs where bacteria survived (i.e., deleted intervals in pair of mutants overlap). Interval graph structure reveals whether DNA is linear or branched DNA. Case for linear genes Lopresti Fall 2007 Lecture

12 Benzer s experiment and interval graphs Case for branched genes Lopresti Fall 2007 Lecture

13 Interval graphs: a comparison Linear genome Branched genome Lopresti Fall 2007 Lecture

14 DNA sequencing: history Sanger method (1977): labeled ddntps* terminate DNA copying at random points. Gilbert method (1977): chemical method to cleave DNA at specific points (G, G+A, T+C, C). Both methods generate labeled fragments of varying lengths that are further electrophoresed. * Dideoxynucleotides, or ddntps, are nucleotides lacking 3'-hydroxyl (-OH) group on their deoxyribose sugar. Note that the sugar normally referred to as deoxyribose lacks a 2'-OH, while the dideoxyribose lacks hydroxyl groups on both its 2' and 3' carbon. The lack of this hydroxyl group means that, after being added by a DNA polymerase to a growing nucleotide chain, no further nucleotides can be added, as no phosphodiester bond can be created. Thus, these molecules form the basis of the dideoxy chaintermination method of DNA sequencing, which was developed by Frederick Sanger in (From Lopresti Fall 2007 Lecture

15 Sanger Method: generating a read Start at primer (restriction site). Grow DNA chain. Include ddntps. Stops reaction at all possible points. Separate products by length, using gel electrophoresis. Lopresti Fall 2007 Lecture

16 DNA sequencing Shear DNA into millions of small fragments. Read nucleotides at a time from the small fragments (Sanger method). Lopresti Fall 2007 Lecture

17 Fragment assembly Computational challenge: assemble individual short fragments (reads) into single genomic sequence. Until late 1990's, the shotgun fragment assembly of human genome was viewed as intractable problem. The Shortest Superstring Problem. Given set of strings, find shortest string containing all of them. Input: Strings s 1, s 2,..., s n. Output: A string s that contains all strings s 1, s 2,..., s n as substrings, such that length of s is minimized. Problem is NP-complete (efficient solution unlikely). This formulation does not take into account sequencing errors! Lopresti Fall 2007 Lecture

18 Shortest Superstring Problem: an example Note: this is an approximation an atttactive mathematical model for a real-world biological problem. Lopresti Fall 2007 Lecture

19 Reducing SSP to TSP Define overlap(s i, s j ) as the length of the longest prefix of s j that matches a suffix of s i. aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa What is overlap(s i, s j ) for these strings? 3? aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa What is overlap(s i, s j ) for these strings? No, 12! Lopresti Fall 2007 Lecture

20 Reducing SSP to TSP Define overlap(s i, s j ) as the length of the longest prefix of s j that matches a suffix of s i. aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa Build graph with n vertices representing strings s 1, s 2,..., s n. Insert edges of length overlap overlap(s i, s j ) between vertices s i and s j. Find longest path which visits every vertex exactly once. This is Traveling Salesman Problem (TSP), which is also NPcomplete. Lopresti Fall 2007 Lecture

21 Reducing SSP to TSP Lopresti Fall 2007 Lecture

22 SSP to TSP: an Example Say that S = {ATC, CCA, CAG, TCC, AGT} SSP AGT CCA ATC ATCCAGT TCC CAG TSP ATC 0 1 AGT 1 2 CAG TCC ATCCAGT CCA Lopresti Fall 2007 Lecture

23 Sequencing by Hybridization (SBH): history 1988: SBH suggested as an an alternative sequencing method. Nobody believed it will ever work. 1991: Light directed polymer synthesis developed by Steve Fodor and colleagues. 1994: Affymetrix develops first 64-kb DNA microarray. First microarray prototype (1989) First commercial DNA microarray prototype w/16,000 features (1994) 500,000 features per chip (2002) Lopresti Fall 2007 Lecture

24 How SBH works Attach all possible DNA probes of length l to flat surface, each probe at distinct and known location. This set of probes is called the DNA array. Apply a solution containing fluorescently labeled DNA fragment to array. The DNA fragment hybridizes with only those probes that are complementary to substrings of length l of fragment. Using a spectroscopic detector, determine which probes hybridize to DNA fragment to obtain l-mer composition of target DNA fragment. Apply combinatorial algorithm (next) to reconstruct sequence of target DNA fragment from l-mer composition. Lopresti Fall 2007 Lecture

25 Hybridization on DNA Array Lopresti Fall 2007 Lecture

26 l-mer composition Spectrum(s, l) is an 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 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} Lopresti Fall 2007 Lecture

27 l-mer composition Spectrum(s, l) is an 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 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} We usually choose lexicographically-ordered representation as canonical one. Lopresti Fall 2007 Lecture

28 Different sequences different spectrums Note that different sequences may have same spectrum. Spectrum(GTATCT, 2)= Spectrum(GTCTAT, 2)= {AT, CT, GT, TA, TC} The Sequencing by Hybridization (SBH) Problem. Reconstruct a string from its l-mer composition. Input: Set S representing all l-mers from an unknown string s. Output: String s such that Spectrum(s, l) = S. Lopresti Fall 2007 Lecture

29 SBH: Hamiltonian Path approach S = {ATG AGG TGC TCC GTC GGT GCA CAG} Corresponding Hamiltonian Path Problem H : ATG AGG TGC TCC GTC GGT GCA CAG ATGCAG GTCC Path visits every vertex once! Lopresti Fall 2007 Lecture

30 SBH: Hamiltonian Path approach A more complicated graph: S = {ATG TGG TGC GTG GGC GCA GCG CGT } H Lopresti Fall 2007 Lecture

31 SBH: Hamiltonian Path approach S = {ATG TGG TGC GTG GGC GCA GCG CGT} Path 1: H Path 2: H ATGCGTGGCA ATGGCGTGCA Lopresti Fall 2007 Lecture

32 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 AT TG GC CA GG Path visits every edge once! Lopresti Fall 2007 Lecture

33 SBH: Eulerian Path approach S = {ATG TGG TGC GTG GGC GCA GCG CGT} corresponds to two different paths: GT CG GT CG 5 4 AT TG 6 4 GC CA AT TG 5 3 GC CA GG ATGGCGTGCA GG ATGCGTGGCA I.e., two acceptable solutions to this SBH problem. Lopresti Fall 2007 Lecture

34 Euler Theorem A graph is balanced if for every vertex, the number of incoming edges equals the number of outgoing edges: in(v) = out(v) Theorem: a connected graph is Eulerian (i.e., has an Euler Cycle) if and only if each of its vertices is balanced. Lopresti Fall 2007 Lecture

35 Euler Theorem: Proof Two cases to consider: (1) Eulerian balanced For every edge entering v (incoming edge) there exists an edge leaving v (outgoing edge). Therefore in(v) = out(v) (2) Balanced Eulerian??? Lopresti Fall 2007 Lecture

36 Algorithm for constructing an Eulerian Cycle (a) Start with an arbitrary vertex v and form arbitrary cycle with unused edges until dead-end is reached. Since graph is Eulerian, dead-end must be starting point, i.e., vertex v. w v Lopresti Fall 2007 Lecture

37 Algorithm for constructing an Eulerian Cycle (b) If cycle from Step (a) earlier is not 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 at starting vertex w. w v Lopresti Fall 2007 Lecture

38 Algorithm for constructing an Eulerian Cycle (c) Combine cycles from Step (a) and Step (b) into single cycle and iterate Step (b). w v Lopresti Fall 2007 Lecture

39 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. I.e., one vertex has an extra outgoing edge, while another corresponding vertex has an extra incoming edge. Lopresti Fall 2007 Lecture

40 Some Difficulties with SBH Fidelity of Hybridization: difficult to detect differences between probes hybridized with perfect matches and those with one or two mismatches. Array Size: effect of low fidelity can be decreased with longer l-mers, but array size increases exponentially in l. Array size is limited with current technology. Practicality: SBH is still impractical. As DNA microarray technology improves, SBH may become practical in future. Practicality (again): although impractical, SBH still spearheaded techniques for expression and SNP analysis. Lopresti Fall 2007 Lecture

41 Traditional DNA sequencing DNA Shake DNA fragments Vector = circular genome (bacterium, plasmid) Known location (restriction site) + = Lopresti Fall 2007 Lecture

42 Different types of vectors VECTOR Plasmid Cosmid BAC (Bacterial Artificial Chromosome) YAC (Yeast Artificial Chromosome) Size of insert (bp) 2,000-10,000 40,000 70, ,000 > 300,000 Not used much recently Lopresti Fall 2007 Lecture

43 Electrophoresis diagrams Lopresti Fall 2007 Lecture

44 Challenging to read sometimes Filtering Smoothening Correcting for length compression Method for deciding nucleotides Lopresti Fall 2007 Lecture

45 Shotgun sequencing Genomic segment Cut many times at random (shotgun) Get one or two reads from each segment ~500 bp ~500 bp Lopresti Fall 2007 Lecture

46 Fragment assembly Reads Cover region with ~7-fold redundancy. Overlap reads, extend to reconstruct original genomic region. Lopresti Fall 2007 Lecture

47 Read coverage C Length of genomic segment: L Number of reads: n Coverage C = (nl) / L Length of each read: l How much coverage is enough? Lander-Waterman model: Assuming uniform distribution of reads, C = 10 results in 1 gapped region per 1,000,000 nucleotides. Lopresti Fall 2007 Lecture

48 Challenges in fragment assembly Repeats: a major problem for fragment assembly. More than 50% of human genome consists of repeats: over 1 million Alu repeats (about 300 bp), about 200,000 LINE repeats (1000 bp and longer). Repeat Repeat Repeat Green and blue fragments are interchangeable when assembling repetitive DNA Lopresti Fall 2007 Lecture

49 Bad things happen when you confuse repeated regions... Correct assembly (we don't know this starting off): Repeat Repeat Repeat Potential mis-assembly (seems just as plausible): Repeat Repeat Repeat Lopresti Fall 2007 Lecture

50 Triazzle: a fun example The puzzle looks simple BUT there are repeats!!! Repeats make it very difficult. Try it only $7.99 at Lopresti Fall 2007 Lecture

51 Repeat types Low-Complexity DNA (e.g., ATATATATACATA ) Microsatellite repeats (a 1 a k ) N where k ~ 3-6 (e.g., CAGCAGTAGCAGCACCAG) Transposons/retrotransposons SINE LINE LTR retroposons Gene Families Short Interspersed Nuclear Elements (e.g., Alu: ~300 bp long, 10 6 copies) Long Interspersed Nuclear Elements ~500-5,000 bp long, 200,000 copies Long Terminal Repeats (~700 bp at each end genes duplicate & then diverge Segmental duplications ~very long, very similar copies Lopresti Fall 2007 Lecture

52 Overlap-Layout-Consensus Assemblers: ARACHNE, PHRAP, CAP, TIGR, CELERA Overlap: find potentially overlapping reads Layout: merge reads into contigs and contigs into supercontigs Consensus: derive the DNA sequence and correct read errors..acgattacaataggtt.. Lopresti Fall 2007 Lecture

53 Overlap Find best match between suffix of one read and prefix of another. Due to sequencing errors, need to use dynamic programming to find optimal overlap alignment. Apply filtration method to filter out pairs of fragments that do not share a significantly long common substring. Lopresti Fall 2007 Lecture

54 Overlapping reads Sort all k-mers in reads (k ~ 24). Find pairs of reads sharing a k-mer. Extend to full alignment throw away if not >95% similar. TACA TAGATTACACAGATTACT GA TAGT TAGATTACACAGATTACTAGA Lopresti Fall 2007 Lecture

55 Overlapping reads and repeats A k-mer that appears N times initiates N 2 comparisons For an Alu that appears 10 6 times, this means comparisons, which is obviously too much. Solution: discard all k-mers that appear more than t x Coverage, (t ~ 10) Lopresti Fall 2007 Lecture

56 Finding overlapping reads Create local multiple alignments from the overlapping reads: TAGATTACACAGATTACTGA TAGATTACACAGATTACTGA TAG TAGATTACACAGATTACTGA TTACACAGATTATTGA TAGATTACACAGATTACTGA TAGATTACACAGATTACTGA TAG TAGATTACACAGATTACTGA TTACACAGATTATTGA Lopresti Fall 2007 Lecture

57 Finding overlapping reads Correct errors using multiple alignment C: 20 C: 35 T: 30 TAGATTACACAGATTACTGA C: 35 TAGATTACACAGATTACTGA C: 40 TAG TAGATTACACAGATTACTGA TTACACAGATTATTGA TAGATTACACAGATTACTGA A: 15 A: 25 - A: 40 A: 25 Score alignments Accept alignments with good scores A: 15 A: 25 A: 0 A: 40 A: 25 C: 20 C: 35 C: 0 C: 35 C: 40 Lopresti Fall 2007 Lecture

58 Layout Repeats are major challenge. Do two aligned fragments really overlap, or are they from two copies of a repeat? Solution: repeat masking hide repeats!!! Masking results in high rate of misassembly (up to 20%). Misassembly means a lot more work at finishing step. Lopresti Fall 2007 Lecture

59 Merge reads into contigs repeat region Merge reads up to potential repeat boundaries Lopresti Fall 2007 Lecture

60 Repeats, errors, and contig lengths Repeats shorter than read length are okay. Repeats with more base pair differences than sequencing error rate are okay. To make smaller portion of genome appear repetitive: - try to increase effective increase read length, - work to decrease sequencing error rate. Lopresti Fall 2007 Lecture

61 Role of error correction Discards ~90% of single-letter sequencing errors. Lowers error rate: decreases effective repeat content, increases contig length. Lopresti Fall 2007 Lecture

62 Merge reads into contigs Repeat region Ignore non-maximal reads. Merge only maximal reads into contigs. Lopresti Fall 2007 Lecture

63 Merge reads into contigs a b Repeat boundary??? Sequencing error Ignore hanging reads, when detecting repeat boundaries. Lopresti Fall 2007 Lecture

64 Merge reads into contigs????? Unambiguous Insert non-maximal reads whenever unambiguous. Lopresti Fall 2007 Lecture

65 Link contigs into supercontigs Normal density Too dense: Overcollapsed? Inconsistent links: Overcollapsed? Lopresti Fall 2007 Lecture

66 Link contigs into supercontigs Find all links between unique contigs. Connect contigs incrementally, if 2 links. Lopresti Fall 2007 Lecture

67 Link contigs into supercontigs Fill gaps in supercontigs with paths of overcollapsed contigs. Lopresti Fall 2007 Lecture

68 Wrap-up Readings for next time: IBP (hash tables, repeat-finding, exact pattern matching, keyword trees, suffix trees). Remember: Come to class having done the readings. Check Blackboard regularly for updates. Lopresti Fall 2007 Lecture