Bioinformatics: Issues and Algorithms Graph Algorithms in

Transcription

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
-2-
The Bridge Obsession Problem
Find a tour crossing every bridge just once.
– Leonhard Euler, 1735
Bridges of Königsberg
-3-
Eulerian Cycle Problem
• Find a cycle that visits
every edge exactly once.
• Algorithm for solution has
linear time complexity.
More complicated Königsberg
-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.
-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
-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)
-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?
-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.
-9-
Benzer’s experiment
Complementation between pairs of mutant T4
bacteriophages
- 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
- 11 -
Benzer’s experiment and interval graphs
Case for branched genes
- 12 -
Interval graphs: a comparison
Linear genome
Branched genome
- 13 -
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 1977. (From
http://encyclopedia.thefreedictionary.com/ddNTP.)
- 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.
- 15 -
DNA sequencing
• Shear DNA into
millions of small
fragments.
• Read 500 – 700
nucleotides at a
time from the small
fragments (Sanger
method).
- 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!
- 17 -
Shortest Superstring Problem: an example
Note: this is an
approximation – an
atttactive mathematical
model for a real-world
biological problem.
- 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
What is overlap(si, sj) for these strings? 3?
What is overlap(si, sj) for these strings? No, 12!
- 19 -
Reducing SSP to TSP
Define overlap(si, sj) as the length of the longest prefix of sj
that matches a suffix of si.
• 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 NPcomplete.
- 20 -
Reducing SSP to TSP
- 21 -
SSP to TSP: an Example
Say that S = {ATC, CCA, CAG, TCC, AGT}
SSP
TSP
AGT
CCA
ATC
ATCCAGT
TCC
CAG
0
1
AGT
2
CAG
1
2
1
1
2
1
2
TCC
ATCCAGT
ATC
- 22 -
CCA
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 commercial
DNA microarray
prototype w/16,000
features (1994)
500,000 features
per chip (2002)
First microarray
prototype (1989)
- 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.
- 24 -
Hybridization on DNA Array
- 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}
- 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.
- 27 -
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.
- 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
ATG CAG GTC C
Path visits every vertex once!
- 29 -
GCA
CAG
A more complicated graph:
S = {ATG TGG TGC GTG GGC GCA GCG CGT }
H
- 30 -
S = {ATG TGG TGC GTG GGC GCA GCG CGT}
Path 1:
H
ATGCGTGGCA
Path 2:
H
ATGGCGTGCA
- 31 -
SBH: Eulerian Path approach
• Vertices correspond to (l–1)-mers:
{AT, TG, GC, GG, GT, CA, CG}
• Edges correspond to l-mers from S.
GT
AT
CG
TG
GC
GG
Path visits every edge once!
CA
- 32 -
SBH: Eulerian Path approach
corresponds to two different paths:
CG
GT
CG
GT
5
6
AT
4
TG
1
4
7
2
5
GC
CA
AT
8
TG
1
3
2
6
GG
3
CA
8
7
GG
ATGGCGTGCA
ATGCGTGGCA
I.e., two acceptable solutions to this SBH problem.
GC
- 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.
- 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
???
- 35 -
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.
- 36 -
v
w
(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.
- 37 -
v
w
(c) Combine cycles from
Step (a) and Step (b) into
single cycle and iterate
Step (b).
- 38 -
v
w
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.
- 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.
- 40 -
Traditional DNA sequencing
DNA
Shake
DNA fragments
Known location
(restriction site)
Vector = circular genome
(bacterium, plasmid)
+
=
- 41 -
Different types of vectors
VECTOR
Size of insert
(bp)
Plasmid
2,000 - 10,000
Cosmid
40,000
BAC (Bacterial Artificial
Chromosome)
70,000 - 300,000
YAC (Yeast Artificial
Chromosome)
> 300,000
Not used much
recently
- 42 -
Electrophoresis diagrams
- 43 -
Challenging to read sometimes
• Filtering
• Correcting for length compression
• Smoothening
• Method for deciding nucleotides
- 44 -
Shotgun sequencing
Genomic segment
Cut many times at
random (shotgun)
Get one or two reads
from each segment
~500 bp
~500 bp
- 45 -
Fragment assembly
Reads
• Cover region with ~7-fold redundancy.
• Overlap reads, extend to reconstruct original genomic region.
- 46 -
Read coverage
C
Length of genomic segment: L
Number of reads:
n
Length of each read:
l
Coverage C = (nl) / 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.
- 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
Green and blue fragments are
interchangeable when
assembling repetitive DNA
- 48 -
Repeat
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
- 49 -
Repeat
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
- 50 -
Repeat types
Low-Complexity DNA
(e.g., ATATATATACATA…)
Microsatellite repeats
(a1…ak)N where k ~ 3-6
(e.g., CAGCAGTAGCAGCACCAG)
Transposons/retrotransposons
SINE
LINE
LTR retroposons
Short Interspersed Nuclear Elements
(e.g., Alu: ~300 bp long, 106 copies)
Long Interspersed Nuclear Elements
~500 - 5,000 bp long, 200,000 copies
Long Terminal Repeats (~700 bp
at each end
Gene Families
genes duplicate & then diverge
Segmental duplications
~very long, very similar copies
- 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
sequence and correct read errors
- 52 -
..ACGATTACAATAGGTT..
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.
- 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 TAGATTACACAGATTAC T GA
|| ||||||||||||||||| | ||
TAGT TAGATTACACAGATTAC TAGA
- 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)
- 55 -
Finding overlapping reads
Create local multiple alignments from the overlapping reads:
TAGATTACACAGATTACTGA
TAG
TTACACAGATTATTGA
TAG
TTACACAGATTATTGA
- 56 -
Finding overlapping reads
• Correct errors using multiple alignment
TAG
TTACACAGATTATTGA
C:
C:
T:
C:
C:
A:
A:
A:
A:
15
25
40
25
• Score alignments
• Accept alignments with good scores
- 57 -
C:
C:
C:
C:
C:
20
35
30
35
40
A:
A:
A:
A:
A:
15
25
0
40
25
20
35
0
35
40
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.
- 58 -
Merge reads into contigs
repeat region
Merge reads up to potential repeat boundaries
- 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.
- 60 -
Role of error correction
Discards ~90% of single-letter sequencing errors.
Lowers error rate:
⇒ decreases effective repeat content,
⇒ increases contig length.
- 61 -
Repeat region
• Ignore non-maximal reads.
• Merge only maximal reads into contigs.
- 62 -
Repeat boundary???
Sequencing
error
b
a
• Ignore “hanging” reads, when detecting repeat boundaries.
- 63 -
?????
Unambiguous
• Insert non-maximal reads whenever unambiguous.
- 64 -
Link contigs into supercontigs
Normal density
Too dense:
Overcollapsed?
Inconsistent links:
Overcollapsed?
- 65 -
Find all links between unique contigs.
Connect contigs incrementally, if ≥ 2 links.
- 66 -
Fill gaps in supercontigs with paths of overcollapsed contigs.
- 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.
- 68 -

Bioinformatics: Issues and Algorithms Graph Algorithms in

Transcription

Similar documents

Broucher

read our latest annual report

Click Here - Excel Group of Institution

Details

ggomeztalk copiakeynote - Sing

Re_ week 5 summary.rtfd