Graph Algorithms in Bioinformatics

Bioinformatics: Issues and Algorithms CSE 308-408 • Fall 2007 • Lecture 13

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 1 - 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

http://www.bioalgorithms.info

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 2 - The Bridge Obsession Problem

Find a tour crossing every bridge just once. – Leonhard Euler, 1735

Bridges of Königsberg

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 3 - Eulerian Cycle Problem

• Find a cycle that visits every edge exactly once. • Algorithm for solution has linear time complexity.

More complicated Königsberg

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 4 - 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.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 5 - 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

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 6 - 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)

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 7 - 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?

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 8 - 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.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 9 - Benzer’s experiment

Complementation between pairs of mutant T4 bacteriophages

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 10 - 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

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 11 - Benzer’s experiment and interval graphs

Case for branched genes

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 12 - Interval graphs: a comparison

Linear genome Branched genome

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 13 - DNA sequencing: history

Sanger method (1977): Gilbert method (1977): labeled ddNTPs* chemical method to terminate DNA copying at cleave DNA at specific random points. 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 chain- termination method of DNA sequencing, which was developed by Frederick Sanger in 1977. (From http://encyclopedia.thefreedictionary.com/ddNTP.) http://www.bioalgorithms.info

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 14 - 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. http://www.bioalgorithms.info

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 15 - DNA sequencing

• Shear DNA into millions of small fragments. • Read 500 – 700 nucleotides at a time from the small fragments (Sanger method).

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 16 - 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 s1, s2, ..., sn.

Output: A string s that contains all strings s1, s2, ..., sn 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!

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 17 - Shortest Superstring Problem: an example

Note: this is an approximation – an atttactive mathematical model for a real-world biological problem.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 18 - Reducing SSP to TSP

Define overlap(si, sj) as the length of the longest prefix of sj

that matches a suffix of si.

aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa

What is overlap(si, sj) for these strings? 3?

aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa

What is overlap(si, sj) for these strings? No, 12!

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 19 - Reducing SSP to TSP

Define overlap(si, sj) as the length of the longest prefix of sj

that matches a suffix of si. aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa aaaggcatcaaatctaaaggcatcaaa

• Build graph with n vertices representing strings s1, s2, ..., sn.

• Insert edges of length overlap overlap(si, sj) between vertices

si and sj. • Find longest path which visits every vertex exactly once. This is Traveling Salesman Problem (TSP), which is also NP- complete.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 20 - Reducing SSP to TSP

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 21 - SSP to TSP: an Example

Say that S = {ATC, CCA, CAG, TCC, AGT}

SSP TSP ATC AGT 0 2 CCA 1 1 AGT 1 CCA ATC 1 ATCCAGT 2 2 2

TCC CAG 1 TCC CAG ATCCAGT

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 22 - Sequencing by Hybridization (SBH): history

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

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 23 - 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. http://www.bioalgorithms.info

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 24 - Hybridization on DNA Array

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 25 - 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}

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 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}

We usually choose lexicographically-ordered representation as canonical one. http://www.bioalgorithms.info

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 27 - Different sequences ⇒ different spectrums

Note that different sequences may have same spectrum.

Spectrum(GTATCT, 2)= {AT, CT, GT, TA, TC} Spectrum(GTCTAT, 2)=

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.

http://www.bioalgorithms.info

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 28 - 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!

http://www.bioalgorithms.info

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 29 - SBH: Hamiltonian Path approach

A more complicated graph:

S = {ATG TGG TGC GTG GGC GCA GCG CGT }

H

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 30 - SBH: Hamiltonian Path approach

S = {ATG TGG TGC GTG GGC GCA GCG CGT}

Path 1:

H

ATGCGTGGCA

Path 2: H

ATGGCGTGCA

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 31 - 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!

http://www.bioalgorithms.info

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 32 - SBH: Eulerian Path approach

S = {ATG TGG TGC GTG GGC GCA GCG CGT} corresponds to two different paths:

GT CG GT CG

5 4 6 5 4 3 AT TG GC CA AT TG GC CA

1 7 8 1 2 8 2 3 6 7 GG GG ATGGCGTGCA ATGCGTGGCA

I.e., two acceptable solutions to this SBH problem.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 33 - 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.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 34 - 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 ???

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 35 - Algorithm for constructing an Eulerian Cycle

(a) Start with an arbitrary vertex v and form arbitrary cycle with v unused edges until dead-end is reached. Since graph is Eulerian, dead-end must be w starting point, i.e., vertex v.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 36 - Algorithm for constructing an Eulerian Cycle

(b) If cycle from Step (a) earlier is not Eulerian cycle, it must v contain a vertex w, which has untraversed edges. Perform Step (a) again, w using vertex w as the starting point. Once again, we will end up at starting vertex w.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 37 - Algorithm for constructing an Eulerian Cycle

(c) Combine cycles from v Step (a) and Step (b) into single cycle and iterate Step (b). w

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 38 - 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.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 39 - 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.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 40 - Traditional DNA sequencing

DNA

Shake

DNA fragments

Known location Vector = circular genome (restriction site) (bacterium, plasmid) + =

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 41 - Different types of vectors

Size of insert VECTOR (bp)

Plasmid 2,000 - 10,000

Cosmid 40,000

BAC (Bacterial Artificial 70,000 - 300,000 Chromosome) > 300,000 YAC (Yeast Artificial Chromosome) Not used much recently

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 42 - Electrophoresis diagrams

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 43 - Challenging to read sometimes

• Filtering • Correcting for length compression • Smoothening • Method for deciding nucleotides

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 44 - Shotgun sequencing

Genomic segment

Cut many times at random (shotgun)

Get one or two reads from each segment

~500 bp ~500 bp

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 45 - Fragment assembly

Reads

• Cover region with ~7-fold redundancy. • Overlap reads, extend to reconstruct original genomic region.

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 46 - 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: Assumssuming uniformorm ddiistribution of reads,eads, C = 10 results in 1 gapped region per 1,1,0000,00,000 nucleeototideses..

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 47 - 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

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 48 - 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

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 49 - Triazzle: a fun example

The puzzle looks simple

BUT there are repeats!!!

Repeats make it very difficult.

Try it – only $7.99 at www.triazzle.com

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 50 - Repeat types

Low-Complexity DNA (e.g., ATATATATACATA…)

N Microsatellite repeats (a1…ak) where k ~ 3-6 (e.g., CAGCAGTAGCAGCACCAG)

Transposons/retrotransposons SINE Short Interspersed Nuclear Elements (e.g., Alu: ~300 bp long, 106 copies) LINE Long Interspersed Nuclear Elements ~500 - 5,000 bp long, 200,000 copies LTR retroposons Long Terminal Repeats (~700 bp at each end

Gene Families genes duplicate & then diverge

Segmental duplications ~very long, very similar copies

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CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 51 - 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 ..ACGATTACAATAGGTT.. sequence and correct read errors

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 52 - 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.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 53 - 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

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 54 - Overlapping reads and repeats

• A k-mer that appears N times initiates N2 comparisons

• For an Alu that appears 106 times, this means 1012 comparisons, which is obviously too much. • Solution: discard all k-mers that appear more than t x Coverage, (t ~ 10)

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 55 - Finding overlapping reads

Create local multiple alignments from the overlapping reads:

TAGATTACACAGATTACTGA TAGATTACACAGATTACTGATAG TTACACAGATTATTGA TAGATTACACAGATTACTGA TAGATTACACAGATTACTGATAG TTACACAGATTATTGA

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 56 - Finding overlapping reads

• Correct errors using multiple alignment

C: 20 C: 20 C: 35 C: 35 T: 30 C: 0 C: 35 C: 35 TAGATTACACAGATTACTGA C: 40 C: 40 TAGTAGATTACACAGATTACTGA TTACACAGATTATTGA TAGATTACACAGATTACTGA A: 15 A: 15 A: 25 A: 25 - A: 0 A: 40 A: 40 A: 25 A: 25 • Score alignments • Accept alignments with good scores

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 57 - 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.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 58 - Merge reads into contigs

repeat region

Merge reads up to potential repeat boundaries

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 59 - 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.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 60 - Role of error correction

Discards ~90% of single-letter sequencing errors. Lowers error rate: ⇒ decreases effective repeat content, ⇒ increases contig length.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 61 - Merge reads into contigs

Repeat region

• Ignore non-maximal reads. • Merge only maximal reads into contigs.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 62 - Merge reads into contigs

Repeat boundary??? Sequencing error b a

• Ignore “hanging” reads, when detecting repeat boundaries.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 63 - Merge reads into contigs

?????

Unambiguous

• Insert non-maximal reads whenever unambiguous.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 64 - Link contigs into supercontigs

Normal density

Too dense: Overcollapsed?

Inconsistent links: Overcollapsed?

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 65 - Link contigs into supercontigs

Find all links between unique contigs.

Connect contigs incrementally, if ≥ 2 links.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 66 - Link contigs into supercontigs

Fill gaps in supercontigs with paths of overcollapsed contigs.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 67 - Wrap-up

Readings for next time: • IBP 9.1-9.5 (hash tables, repeat-finding, exact pattern matching, keyword trees, suffix trees).

Remember: • Come to class having done the readings. • Check Blackboard regularly for updates.

CSE 308-408 · Bioinformatics: Issues and Algorithms Lopresti · Fall 2007 · Lecture 13 - 68 -