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• Introduction to Graph Theory• Eulerian & Hamiltonian Cycle Problems• Benzer Experiment and Interval Graphs • DNA Sequencing• The Shortest Superstring & Traveling
Salesman Problems• Sequencing by Hybridization • Fragment Assembly and Repeats in DNA • Fragment Assembly Algorithms
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Bridges of Königsberg
Find a tour crossing every bridge just onceLeonhard Euler, 1735
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• Find a cycle that visits every edge exactly once
• Linear time
More complicated Königsberg
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• Find a cycle that visits every vertexexactly once
• NP – complete
Game invented by Sir William Hamilton in 1857
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• Arthur Cayley studied chemical structures of hydrocarbons in the mid-1800s
• He used trees(acyclic connected graphs) to enumerate structural isomers
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Benzer’s work• Developed deletion
mapping• “Proved” linearity of
the gene• Demonstrated
internal structure of the gene
Seymour Benzer, 1950s
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• Normally bacteriophage T4 kills bacteria • However if T4 is mutated (e.g., an important gene is
deleted) it gets disabled and looses the ability to kill bacteria
• Suppose the bacteria is infected with two different mutants each of which is disabled – would the bacteria still survive?
• Amazingly, a pair of disabled viruses can kill a bacteria even if each of them is disabled.
• How can it be explained?
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• Idea: infect bacteria with pairs of mutant T4 bacteriophage (virus)
• Each T4 mutant has an unknown interval deleted from its genome
• If the two intervals overlap: T4 pair is missing part of its genome and is disabled – bacteria survive
• If the two intervals do not overlap: T4 pair has its entire genome and is enabled –bacteria die
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Complementation between pairs of mutant T4 bacteriophages
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• Construct an interval graph : each T4 mutant is a vertex, place an edge between mutant pairs where bacteria survived (i.e., the deleted intervals in the pair of mutants overlap)
• Interval graph structure reveals whether DNA is linear or branched DNA
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Linear genome Branched genome
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Sanger method (1977): labeled ddNTPsterminate DNA copying at random points.
Both methods generate labeled fragments of varying lengths that are further electrophoresed.
Gilbert method (1977):chemical method to cleave DNA at specific points (G, G+A, T+C, C).
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1. Start at primer(restriction site)
2. Grow DNA chain
3. Include ddNTPs
4. Stops reaction at all possible points
5. Separate products by length, using gel electrophoresis
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• Shear DNA into millions of small fragments
• Read 500 – 700 nucleotides at a time from the small fragments (Sanger method)
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• Computational Challenge : assemble individual short fragments (reads) into a single genomic sequence (“superstring”)
• Until late 1990s the shotgun fragment assembly of human genome was viewed as intractable problem
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• Problem: Given a set of strings, find a shortest string that contains all of them
• Input: Strings s1, s2,…., sn• Output: A string s that contains all strings
s1, s2,…., sn as substrings, such that the length of s is minimized
• Complexity: NP – complete • Note: this formulation does not take into
account sequencing errors
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Shortest Superstring Problem: Example
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• 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?
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• Define overlap ( si, sj ) as the length of the longest prefix of sj that matches a suffix of si.aaaggcatcaaatctaaaggcatcaaa
aaaggcatcaaatctaaaggcatcaaaaaaggcatcaaatctaaaggcatcaaa
overlap=12
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• Define overlap ( si, sj ) as the length of the longest prefix of sj that matches a suffix of si.aaaggcatcaaatctaaaggcatcaaa
aaaggcatcaaatctaaaggcatcaaaaaaggcatcaaatctaaaggcatcaaa
• Construct a graph with n vertices representing the n strings s1, s2,…., sn.
• Insert edges of length overlap ( si, sj ) between vertices siand sj.
• Find the shortest path which visits every vertex exactly once. This is the Traveling Salesman Problem (TSP), which is also NP – complete.
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(cont’d)
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S = { ATC, CCA, CAG, TCC, AGT }
SSPAGT
CCA
ATC
ATCCAGT
TCC
CAG
ATCCAGT
TSP ATC
CCA
TCC
AGT
CAG
2
2 22
1
1
10
11
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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 commercialDNA microarrayprototype w/16,000features (1994)
500,000 featuresper chip (2002)
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• Attach all possible DNA probes of length l to a flat surface, each probe at a distinct and known location. This set of probes is called the DNA array.
• Apply a solution containing fluorescently labeled DNA fragment to the array.
• The DNA fragment hybridizes with those probes that are complementary to substrings of length lof the fragment.
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(cont’d)
• Using a spectroscopic detector, determine which probes hybridize to the DNA fragment to obtain the l–mer composition of the target DNA fragment.
• Apply the combinatorial algorithm (below) to reconstruct the sequence of the target DNA fragment from the l – mer composition.
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l• Spectrum ( s, l ) - 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}
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l• Spectrum ( s, l ) - 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}
• We usually choose the lexicographically maximal representation as the canonical one.
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Different sequences – the same spectrum
• Different sequences may have the same spectrum:
Spectrum(GTATCT,2)=Spectrum(GTCTAT,2)={AT, CT, GT, TA, TC}
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• Goal: Reconstruct a string from its l-mercomposition
• Input: A set S, representing all l-mers from an (unknown) string s
• Output: String s such that Spectrum ( s,l ) = S
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S = { ATG AGG TGC TCC GTC GGT GCA CAG }
Path visited every VERTEX once
ATG AGG TGC TCC GTC GGT GCA CAG
ATGCAGGTCC
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A more complicated graph:
S = { ATG TGG TGC GTG GGC GCA GCG CGT }
HH
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S = { ATG TGG TGC GTG GGC GCA GCG CGT }
Path 1:
HH
ATGCGTGGCA
HH
ATGGCGTGCA
Path 2:
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S = { ATG, 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
AT
GT CG
CAGCTG
GGPath visited every EDGE once
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S = { AT, TG, GC, GG, GT, CA, CG } corresponds to two different paths:
ATGGCGTGCA ATGCGTGGCA
AT TG GCCA
GG
GT CG
AT
GT CG
CAGCTG
GG
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• 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.
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• Eulerian → balanced
for every edge entering v (incoming edge) there exists an edge leaving v (outgoing edge). Therefore
in(v)=out(v)
• Balanced → Eulerian
???
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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.
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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 untraversededges. Perform step (a) again, using vertex w as the starting point. Once again, we will end up in the starting vertex w.
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Algorithm for Constructing an Eulerian Cycle (cont’d)
c. Combine the cycles from (a) and (b) into a single cycle and iterate step (b).
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• 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.
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• Fidelity of Hybridization: difficult to detect differences between probes hybridized with perfect matches and 1 or 2 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 the future
• Practicality again : Although SBH is still impractical, it spearheaded expression analysis and SNP analysis techniques
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DNA
Shake
DNA fragments
VectorCircular genome(bacterium, plasmid)
Knownlocation(restrictionsite)
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> 300,000Not used much
recently
YAC (Yeast Artificial Chromosome)
70,000 - 300,000BAC (Bacterial Artificial
Chromosome)
40,000Cosmid
2,000 - 10,000 Plasmid
Size of insert (bp)VECTOR
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• Filtering
• Smoothening
• Correction for length compressions
• A method for calling the nucleotides – PHRED
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cut many times at random (Shotgun)
genomic segment
Get one or two reads from each
segment~500 bp ~500 bp
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Cover region with ~7-fold redundancyOverlap reads and extend to reconstruct the
original genomic region
reads
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Length of genomic segment: L
Number of reads: n Coverage C = n l / LLength 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
CCCC
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• Repeats: A major problem for fragment assembly• > 50% of human genome are 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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The puzzle looks simple
BUT there are repeats!!!
The repeats make it very difficult.
Try it – only $7.99 atwww.triazzle.com
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• Low-Complexity DNA (e.g. ATATATATACATA…)
• Microsatellite repeats (a1…ak)N 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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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..
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• Find the best match between the suffix of one read and the prefix of another
• Due to sequencing errors, need to use dynamic programming to find the optimal overlap alignment
• Apply a filtration method to filter out pairs of fragments that do not share a significantly long common substring
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TAGATTACACAGATTAC
TAGATTACACAGATTAC
|||||||||||||||||
• 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
T GA
TAGA
| ||TACA
TAGT
||
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• A k-mer that appears N times, initiates N2
comparisons
• For an Alu that appears 106 times � 1012
comparisons – too much
• Solution:Discard all k-mers that appear more than
t × Coverage, (t ~ 10)
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Create local multiple alignments from the overlapping reads
TAGATTACACAGATTACTGATAGATTACACAGATTACTGATAG TTACACAGATTATTGATAGATTACACAGATTACTGATAGATTACACAGATTACTGATAGATTACACAGATTACTGATAG TTACACAGATTATTGATAGATTACACAGATTACTGA
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• Repeats are a major challenge• Do two aligned fragments really overlap, or
are they from two copies of a repeat? • Solution: repeat masking – hide the
repeats!!!• Masking results in high rate of misassembly
(up to 20%)• Misassembly means alot more work at the
finishing step
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Merge reads up to potential repeat boundaries
repeat region
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Repeats, Errors, and Contig Lengths
• Repeats shorter than read length are OK
• Repeats with more base pair differencessthan sequencing error rate are OK
• To make a smaller portion of the genome appear repetitive, try to:• Increase read length• Decrease sequencing error rate
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Too dense: Overcollapsed?
Inconsistent links: Overcollapsed?
Normal density
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• A consensus sequence is derived from a profile of the assembled fragments
• A sufficient number of reads is required to ensure a statistically significant consensus
• Reading errors are corrected
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Derive multiple alignment from pairwise read alignments
TAGATTACACAGATTACTGA TTGATGGCGTAA CTATAGATTACACAGATTACTGACTTGATGGCGTAAACTATAG TTACACAGATTATTGACTTCATGGCGTAA CTATAGATTACACAGATTACTGACTTGATGGCGTAA CTATAGATTACACAGATTACTGACTTGATGGGGTAA CTA
TAGATTACACAGATTACTGACTTGATGGCGTAA CTA
Derive each consensus base by weighted voting
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EULER - A New Approach to Fragment Assembly
• Traditional “overlap-layout-consensus” technique has a high rate of mis-assembly
• EULER uses the Eulerian Path approach borrowed from the SBH problem
• Fragment assembly without repeat masking can be done in linear time with greater accuracy
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Repeat Repeat Repeat
Find a path visiting every VERTEX exactly once: Hamiltonian path problem
Each vertex represents a read from the original sequence.Vertices from repeats are connected to many others.
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Repeat Repeat Repeat
Find a path visiting every EDGE exactly once:Eulerian path problem
Placing each repeat edge together gives a clear progression of the path through the entire sequence.
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Repeat1 Repeat1Repeat2 Repeat2
Can be easily constructed with any number of repeats
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• Construction of repeat graph from k – mers: emulates an SBH experiment with a huge (virtual) DNA chip.
• Breaking reads into k – mers: Transform sequencing data into virtual DNA chip data.
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(cont’d)
• Error correction in reads: “consensus first”approach to fragment assembly. Makes reads (almost) error-free BEFORE the assembly even starts.
• Using reads and mate-pairs to simplify the repeat graph (Eulerian Superpath Problem).
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Find a path visiting every VERTEX exactly once in the OVERLAP graph:
Hamiltonian path problem
NP-complete: algorithms unknown
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(cont’d)
Find a path visiting every EDGE exactly once in the REPEAT graph:
Eulerian path problem
Linear time algorithms are known
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• Problem: Construct the repeat graph from a collection of reads.
• Solution: Break the reads into smaller pieces.
?
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• Virtual DNA chip allows the biological problem to be solved within the technological constraints.
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(cont’d)
• Reads are constructed from an original sequence in lengths that allow biologists a high level of certainty.
• They are then broken again to allow the technology to sequence each within a reasonable array.
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• If an error exists in one of the 20-mer reads, the error will be perpetuated among all of the smaller pieces broken from that read.
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(cont’d)
• However, that error will not be present in the other instances of the 20-mer read.
• So it is possible to eliminate most point mutation errors before reconstructing the original sequence.
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• Graph theory is a vital tool for solving biological problems
• Wide range of applications, including sequencing, motif finding, protein networks, and many more
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• Simons, Robert W. Advanced Molecular Genetics Course, UCLA (2002). http://www.mimg.ucla.edu/bobs/C159/Presentations/Benzer.pdf
• Batzoglou, S. Computational Genomics Course, Stanford University (2004). http://www.stanford.edu/class/cs262/handouts.html