Download The Principles of Shotgun Sequencing and Automated Fragment

The Principles of Shotgun Sequencing and
Automated Fragment Assembly
Special excerpt for lecture
c
°Martti
T. Tammi
[email protected]
Center for Genomics and Bioinformatics,
Karolinska Institutet,
Stockholm, Sweden
April 13, 2003
1
Contents
1 Introduction
1.1
1.2
1.3
3
The Human Genome Project . . . . . . . . . . . . . . . . . . . . .
3
1.1.1
The First Shotgun Projects . . . . . . . . . . . . . . . . . .
4
The Principle of Shotgun Sequencing . . . . . . . . . . . . . . . . .
5
1.2.1
Strategies for Sequencing . . . . . . . . . . . . . . . . . . .
5
1.2.2
The Hierarchical Sequencing (HS) Strategy or Clone by Clone
Sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2.3
The Whole Genome Shotgun (WGS) Strategy . . . . . . . .
7
1.2.4
Combining HS and WGS . . . . . . . . . . . . . . . . . . .
7
1.2.5
Shotgun Sequencing Methods . . . . . . . . . . . . . . . . .
8
Development of Shotgun Fragment Assembly Tools . . . . . . . . .
9
2 Milestones in Genome Sequencing
11
2.0.1
First Cellular Organisms
. . . . . . . . . . . . . . . . . . .
11
2.0.2
First Multicellular Organisms . . . . . . . . . . . . . . . . .
11
3 Shotgun Fragment Assembly Algorithms
12
3.1
Types of Problems in Sequence Assembly . . . . . . . . . . . . . .
12
3.2
Complications in Fragment Assembly . . . . . . . . . . . . . . . . .
13
3.3
Base calling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.4
The Data Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.5
Error Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.6
Genome Coverage
. . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.7
Distance Constraints . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.8
Fragment Assembly Problem Models . . . . . . . . . . . . . . . . .
22
3.8.1
The Shortest Common Superstring Model . . . . . . . . . .
24
3.8.2
The Sequence Reconstruction Model . . . . . . . . . . . . .
25
3.8.3
Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
3.9.1
28
3.6.1
3.9
Contamination . . . . . . . . . . . . . . . . . . . . . . . . .
1
3.9.2
Overlap Detection. . . . . . . . . . . . . . . . . . . . . . . .
28
3.9.3
Contig Layout Determination.
. . . . . . . . . . . . . . . .
29
3.9.4
Creation of Multiple Alignments. . . . . . . . . . . . . . . .
30
3.9.5
Computation of the Consensus Sequence. . . . . . . . . . .
30
4 Shotgun Fragment Assembly Programs
4.1
31
Phrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.1.1
The Phrap Method . . . . . . . . . . . . . . . . . . . . . . .
32
4.1.2
Greedy Algorithm for Layout Construction . . . . . . . . .
33
4.1.3
Construction of Contig Sequences . . . . . . . . . . . . . . .
34
2
1
Introduction
The secret of life is complementarity. Deoxyribonucleic acid, DNA, is a double
helix consisting of alternating phosphate and sugar groups. The two chains are
held together by hydrogen bonds between pairs of organic bases adenine (A) with
thymine (T) and guanine (G) with cytosine (C) (Watson & Crick, 1953). In 1962
James Watson, Francis Crick, and Maurice Wilkins jointly received The Nobel
Prize in Physiology or Medicine for their determination of the structure of DNA.
In 1959, Seymour Benzer presented the idea that DNA and RNA molecules can be
seen as linear words over a 4-letter alphabet (Benzer, 1959). It is the order of these
letters (bases) that codes for the important genetic information of an organism.
DNA can be thought of as the language of life, the four letters in an arrangement
of words, each word consisting of three letters, codons, that code for amino acids.
The codons or words are arranged in sentences that code for specific proteins.
The language of DNA dictates which proteins are made, as well as when and
how much. In the early days the determination of the nucleotide sequence was a
strenuous task and computer programs were implemented to aid in the sequencing
process. These, and in continuation attempts to decipher the language of DNA
were the seeds that led to the development of the field of bioinformatics. Today
vast amounts of data are available for analysis in various databases around the
world. The rapid development of bioinformatics was due to the fast development
of computer hardware and the need of large genome projects to store, organize
and analyze increasing amounts of data. Genome projects were in turn facilitated
by the development of improved bioinformatics tools.
1.1
The Human Genome Project
The development of computational solutions to problems in genomics and sequence
handling have been crucial to the decision to undertake and successfully carry out
the ambitious effort to sequence the whole human genome, approximately three
billion base pairs, and the sequencing of many other genomes. The Human Genome
Project (HGP) began formally in October 1990 when the Department of Energy
(DOE) and National Institute of Health (NIH) presented a joint five-year plan to
the U.S. Congress. The document 1 says, ”improvements in technology for almost
every aspect of genomics research have taken place. As a result, more specific
goals can now be set for the project.”. This international effort was originally
estimated to require 15 years and $3 billion. This plan was revised in 1998, when
J. Craig Venter, PE Corporation and The Institute of Genomic Research (TIGR)
proposed to launch a joint venture that would sequence the entire human genome
in three years (Marshall, 1999). In June of 2000, the publicly sponsored Human
Genome Project leaders, the private company Celera Genomics, the U.S. President
Bill Clinton, and the Prime Minister of Great Britain Tony Blair announced the
completion of a ”working draft” sequence of the human genome. A special issue
1 Coauthored by DOE and NIH and titled Understanding Our Genetic Inheritance, The U.S.
Human Genome Project: The First Five Years (FY 1991-1995)
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of Nature, Feb. 15, 2001 contains the initial analyses of the draft sequence of
the International Human Genome Sequencing Consortium, while a special issue
of Science, Feb. 16, 2001, reported the draft sequence reported by the private
company, Celera Genomics. The draft sequence is to be improved to achieve a
complete, high-quality DNA reference sequence by 2003, two years earlier than
originally projected.
1.1.1
The First Shotgun Projects
In 1965, Robert William Holley and co-workers published the complete sequence
of the alanine tRNA from yeast (Holley et al., 1965), a task that took seven years
to complete. Since the sequence of only small fragments could be determined, a
fragmentation stratagem had to be used. An enzyme obtained from snake venom
was used to remove nucleotides one by one from a fragment, to produce a mixture of smaller fragments of all possible intermediate lengths. This mixture was
subsequently separated by passing it through a chromatography column. By the
determining the identity of the terminal nucleotide of each fraction and the knowledge of its length, it was possible to establish the sequence of the smaller fragments,
as well as the sequence of the original large fragment using basic overlapping logic.
The first three shotgun sequencing projects were the 6,569 base pair human mitochondrial DNA (Anderson et al., 1981), the bacteriophage λ nucleotide sequence of
48,502 base pairs (Sanger et al., 1982) and the Epstein-Barr virus at 172,282 base
pairs (Baer et al., 1984). In the shotgun sequencing method (Anderson, 1981;
Messing et al., 1981; Deininger, 1983) the sequence is broken into small pieces,
since the length of the longest contiguous DNA stretch that can be determined is
limited by the length that can be read using an electrophoresis gel. In the bacteriophage λ project, the whole λ genome was sonicated to yield small fragments,
(Fuhrman et al., 1981) in order to create a random library of fragmented DNA.
These fragments were cloned using a single stranded bacteriophage M13 (Gronenborn & Messing, 1978; Messing et al., 1981) as a vector. This cloning procedure
replaced the previous methods. All the sequences were assembled using Rodger
Stadens assembly program (Staden, 1982a; Staden, 1982b).
After the introduction of gel electrophoresis based sequencing methods in 1977 by
Maxam & Gilbert using chemical degradation of DNA (Maxam & Gilbert, 1977)
and chain termination techniques by Frederick Sanger and his colleagues (Sanger
et al., 1977), it became possible to determine longer sequences. These techniques
allow a routine determination of approximately 300 consecutive bases of a DNA
strand in one reaction.
Until the advent of fluorescent sequencing, polyacrylamide slab gel electrophoresis
was used together with radioactive 32 P, 33 P or 35 S labels and autoradiographs.
Fluorescent labelling, i.e. dyes that can be detected after excitation using a laser,
allowed automatic detection of the base sequence (Smith et al., 1986). Regardless
of refinements of the sequencing methods, the average fragment length has not
grown significantly. In some conditions and using special care it is possible to
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produce reads that are over 1000 bases long, but only average lengths of about
400 to 800 base pairs are feasible in routine production.
Attempts to increase the sequencing speed and throughput, as well as attempts to
lower the cost has led to the development of alternative strategies, such as sequencing by hybridization (Drmanac et al., 1989), Massively Parallel Signature Sequencing (MPSS) (Brenner et al., 2000a; Brenner et al., 2000b), Matrix-assisted Laser
Desorption Ionization Time-of-Flight (MALDI-TOF-MS) (Karas & Hillenkamp,
1988), Pyrosequencing (Ronaghi & Nyren, 1998) and the use of Scanning Tunneling Microscopy (Allison et al., 1990). However, none of these methods have as
yet contributed to large-scale genomic sequencing due to their short read lengths,
but many of them are well suited for e.g. Single Nucleotide Polymorphism (SNP)
genotyping. Thus, Sanger sequencing remains the method of choice for large-scale
sequencing.
1.2
The Principle of Shotgun Sequencing
Although the computational problems involved in fragment assembly have been
studied almost 30 years, they still continue to receive attention, mainly because of
large sequencing projects, such as human genome project and many other projects,
necessitates access to efficient algorithms for precise assembly of long DNA sequences and attempts to sequence whole genomes, using the whole genome shotgun sequencing approach revealed some limitations in currently used methods.
Several genomes have been published as drafts, but none of the eukaryote genomes
are completely finished, yet. This is mainly due to ubiquitous repeated sequences
that complicate the fragment assembly and finishing.
The core of the problem is that it is not routinely possible to sequence larger
contiguous fragments than approximately 500 to 800 bases. In some conditions
and much work it may be possible to get sequence over 1 kb in length, but this
is not feasible in normal, automated production. Genomes are much longer than
this. The length of human genome is about 3 Gb. X chromosome is about 150 Mb
and the human mitochondrion genome alone is 16.5 kb. To determine the base
sequence of long DNA stretches we obviously need some stratagem. The shotgun
sequencing method has proven to be the method of choice as the solution to this
problem. DNA is first cut into small pieces, each of these fragments are sequenced
individually and then puzzled together to create the original contiguous sequence,
contig. The advantages of the shotgun sequencing method are easy automation
and scalability.
1.2.1
Strategies for Sequencing
In one approach, the whole genome of an organism is broken into small pieces,
which are then sequenced, often from both ends. This approach is called Whole
Genome Shotgun Sequencing (WGS). Another approach to sequence a genome is
more hierarchical. The genome is first broken into more manageable pieces of approximately 100 to 200 kb, which are inserted into bacterial artificial chromosomes,
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BACs, to yield a library. The BACs are ordered by Sequence Tagged Sites (STS)
mapping or by fingerprinting to result in a minimally overlapping tiling path, that
covers the whole genome. The BACs are in turn sequenced using the shotgun
approach. Thus, the shotgun method applies also to the Hierarchical Sequencing
(HS) Strategy. The most powerful method may be the combination of the two
approaches (see below).
A physical map is a set of DNA fragments whose relative positions in the genome
are known. The ultimate physical map is the complete sequence of a genome.
Clones in a library can be ordered either by fingerprinting, hybridization or endsequencing. The presence of significant repeat content may cause a failure to provide an unique link between segments of a genome. Further, the stochastic method
to sample a library often results in certain segments being over-represented, underrepresented or to be completely absent, due to experimental procedures. For this
reason, it is essential to use more than one library and more than one cloning
system.
1.2.2
The Hierarchical Sequencing (HS) Strategy or Clone by Clone
Sequencing
The HS strategy involves an initial mapping step. A physical map is built using
clones with large inserts, such as BACs. The minimum tiling path of clones is
selected to cover the whole genome. Each clone is individually sequenced using
a shotgun strategy. A mixture of clone types is usually used in sequencing large
genomes, also the different chromosomes may be separated and used to create
chromosome specific libraries.
This hierarchical strategy has been used for example for bacterial genomes, the
nematode C. elegans(C. Elegans Sequencing Consortium, 1998) and the yeast, S.
cerevisiae (Goffeau et al., 1996), A. thaliana (The Arabidopsis Genome Initiative,
2000), in a currently ongoing P. falciparum, the human malaria parasite project
(P. falciparum Genome Sequencing Consortium, 1996), and also in sequencing the
whole human genome by the International Human Genome Sequencing Consortium.
The physical map can also be built as the sequencing progresses. This is called the
“map as you go” strategy. Venter et al. developed an efficient method for selecting
minimally overlapping clones by sequence-tagged connectors (STCs) (Venter et al.,
1996), which is commonly referred to as ”BAC-end sequencing”. The end of
clones from a large insert library are sequenced, yielding sequence tags. Once a
selected clone is sequenced, these tags can be used to select a flanking, minimally
overlapping clone to be subsequently sequenced. When no overlapping clone exist,
a new seed point is selected.
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1.2.3
The Whole Genome Shotgun (WGS) Strategy
In the whole genome shotgun strategy the genome is fragmented randomly without
a prior mapping step. The computer assembly of sequence reads is more demanding
than in the HS strategy, due to the lacking positional information. Therefore, the
use of varied clone lengths and the sequencing of both ends is essential to build
scaffolds (Edwards & Caskey, 1990; Chen et al., 1993; Smith et al., 1994; Kupfer
et al., 1995; Roach et al., 1995; Nurminsky & Hartl, 1996; Roach et al., 1995),
in particular when sequencing large, complex genomes. This strategy has been
used to sequence e.g. Haemophilus influenzae Rd. (Fleischmann et al., 1995), the
580,070 bp genome of Mycoplasma genitalium, the smallest known genome of any
free-living organism and the Drosophila melanogaster genome (Adams et al., 2000;
Myers et al., 2000). However, a clone-by-clone based strategy has been used to
finish the Drosophila genome.
The WGS approach to sequence the human genome was proposed (Weber & Myers,
1997). This started a debate of the feasibility of the WGS strategy to sequence
the entire human genome (Green, 1997; Eichler, 1998; The Sanger Centre, 1998;
Waterston et al., 2002).
The method was applied to the whole human genome by the private company
Celera Genomics.
1.2.4
Combining HS and WGS
In some cases when one considers the project type, cost and time involved, it may
be advantageous to combine the WGS and HS strategy. Mathematical models are
helpful aids to determine an applicable sequencing approach. Several mathematical
models have been developed and simulations performed for a variety of sequencing
strategies (Clarke & Carbon, 1976; Lander & Waterman, 1988; Idury & Waterman,
1995; Schbath, 1997; Roach et al., 2000; Batzoglou et al., 1999). These models are
useful in conjunction with large genome sequencing projects. The models allow
the monitoring of the progress of the project to control deviations from predicted
values, such as coverage, that may arise due to biological or technical problems,
and thereby aid in taking early measures to adjust the problem. These models are
also advantageous in planning a sequencing project, estimating project duration
and allocating resources, as well as optimizing cost.
A rearrangement (Anderson, 1981) of the Clarke and Carbon expression (Clarke
& Carbon, 1976) can be used to monitor how random sequencing progresses:
³
r ´n
fl = 1 − 1 −
L
The equation gives the fraction, fl of the target sequence, G of length L in contigs,
when n number of reads of the length r are sequenced.
Lander and Waterman developed a mathematical model for genomic mapping by
fingerprinting random clones (Lander & Waterman, 1988), which can be considered
7
analogous to the well-known Clarke-Carbon formulas. The Lander and Waterman
expression can be used to calculate the average number of gaps, contigs and their
average length in a shotgun project; a target sequence G is redundantly sampled,
at an average coverage c̄ = nr̄/L and assuming that the fragments are uniformly
distributed along the target sequence, the coverage at a given base b is a Poisson
random variable with mean c̄:
P(base b covered by k f ragments) =
e−c̄ · c̄k
k!
(1)
For example, the fraction of G that is covered by at least one fragment is 1 − e−c̄ .
Comparison analysis of map-based sequencing to WGS (Wendl et al., 2001) show
that, regardless of library type, random selection initially yields comparable rates
of coverage and lower redundancy than map-based sequencing. This advantage
shifts to the map-based approach after a certain number of clones have been sequenced. The map-based sequencing essentially yields a linear coverage increase,
if a constant redundancy can be maintained, as opposed to the asymptotic process
of random clone selection, equation 1.
Considering the number of clones needed with each method to achieve a certain
coverage, it is reasonable to combine the approaches so that a random strategy
is applied early in the project but finished by a map-based approach. The advantages and disadvantages of each method to sequence the human genome has
been extensively debated in the literature (Weber & Myers, 1997; Green, 1997;
Waterston et al., 2002).
The advantage of the hierarchical approach is that it minimizes large misassmeblies, since each clone is separately assembled. On the other hand, the WGS
method eliminates the time consuming mapping step and relies largely on the performance of fragment assembly algorithms to solve larger and more complicated
assembly problems. For example, the first five years of the public human genome
project were spent in mapping (National Research Counsil, 1988; Watson, 1990).
1.2.5
Shotgun Sequencing Methods
Both HS and WGS involve shotgun sequencing. In the shotgun process, the target
sequence is broken into small pieces as randomly as possible. The fractionation
is usually performed by passing the fragments through a nozzle under regulated
pressure, for example by nebulization, by sonication or by partial digest with
restriction enzymes. These processes normally yield a semi-random collection of
fragments. In order to remove fragments that are too long or too short, the
fragments are loaded onto an agarose gel and separated by electrophoresis. The
section of the gel containing the desired size is cut out and purified to yield a set
of fragments of similar size. Since there are many copies of the original target
sequence, many of these fragments overlap.
To get a sufficient quantity of each unique fragment, these size selected fragments
are inserted into a cloning vector, mostly using a ligation reaction. The vectors
8
are inserted into ’competent’ E. coli cells, and as the colonies grow, the insert is
effectively amplified. A sufficient number of colonies or plaques each containing a
separate clone, are selected to achieve the desired coverage of the target sequence,
usually about ten-fold.
The vectors used (M13 or plasmid) contain primer sites for reverse and forward
primers at the region where the restriction enzyme cuts, to allow sequencing from
each end of the insert. Sequencing from both ends yields additional positional
information, since the distance by which the read pair is separated is known (Edwards & Caskey, 1990). After a subsequent purification step to remove all bacterial
components, media, proteins and other contaminants, the fragments are sequenced
using dye primer or dye terminator techniques. The sequence is read using an automatic sequencer, where electropherograms or chromatograms containing the signal
sequence read from the sequencing gel are created. The actual determination of
the base sequence, the base-calling, is based on this signal sequence. The resulting
reads are assembled in a computer to reconstruct the original target sequence. Due
to cloning biases and systematic failures in the sequencing chemistry, the random
data alone is usually insufficient to yield a complete sequence. Thus, the computer
assembly step results in several contigous sequences, contigs (Staden, 1980). In
order to obtain the complete sequence, the remaining gaps must be filled and remaining problems solved. This can be accomplished by directed sequencing after
the contigs have been ordered. This finishing process is usually a time consuming
and labor intensive step.
1.3
Development of Shotgun Fragment Assembly Tools
In the 1960’s, several theories and computer programs were developed to aid sequence reconstruction based on the overlap method (Dayhoff, 1964; Shapiro et al.,
1965; Mosimann et al., 1966; Shapiro, 1967). George Hutchinson at National Institutes of Health (NIH) was the first to apply the mathematical theory of graphs
to reconstruct an unknown sequence from small fragments (Hutchinson, 1969).
The new electrophoresis technique devised by Sanger et al. allowed faster sequence
determination, which in turn resulted in an increased need for computer programs
to handle the larger amount of sequence data. During the years 1977 to 1982,
Rodger Staden published a number of papers on constructing sequence fragment
assembly and project management software (Staden, 1977; Staden, 1978; Staden,
1979; Staden, 1980; Staden, 1982a; Staden, 1982b). In Staden’s original algorithm,
the shotgun fragments were overlapped and merged in a sequential manner, so that
in each step, a fragment chosen from a database was tested against all merged
fragments.
Alignment techniques for sequences were devised in parallel with the development
of fragment assembly programs (Needleman & Wunsch, 1970; Smith & Waterman, 1981). These methods allowed comparison and optimal alignment between
sequences with insertions and deletions, corresponding to e.g. sequencing errors.
The errors in sequence reads are dispersed, but in general the number of errors
9
increases toward the ends of reads, due to the geometric distribution of the difference between the growing sizes of fragments. Despite the erroneous input data,
all the early fragment assembly models were based on the unrealistic assumption
that sequence reads are free of errors. Peltola et al. were the first to define the
sequence assembly problem involving errors (Peltola et al., 1983) and also developed the SEQAID program based on their model (Peltola et al., 1984). Earlier
programs and models were merely ad hoc or based on models, that allowed no
errors in the shotgun data e.g. (Smetanič & Polozov, 1979; Gingeras et al., 1979;
Gallant, 1983). In 1994, Mark J. Miller and John I. Powell compared 11 early
sequence assembly programs for the accuracy and reproducibility with which they
assemble fragments. Only three of the programs, consistently produced consensus
sequences of low error rate and high reproducibility, when tested on a sequence
that contained no repeats (Miller & Powell, 1994).
The general outline of most assembly algorithms is: 1. Examine all pairs and
create a set of candidate overlaps, 2. Form an approximate layout of fragments,
3. Make multiple alignments of the layout, 4. Create the consensus sequence, see
section 3.9. All existing methods rely on heuristics, since the fragment assembly
problem is NP-hard (section 3.1) (Garey & Johnson, 1979; Gallant et al., 1980;
Gallant, 1983; Kececioglu, 1991). Hence, it is not possible to exactly compute
the optimal solution if the amount of input data is large enough (section 3.1).
Kececioglu and Myers made a great deal of progress by identifying the optimality
criteria and developing methods to solve them (Kececioglu, 1991; Kececioglu &
Myers, 1995).
In 1990, a pairwise end-sequencing strategy was described (Edwards & Caskey,
1990). This was followed by the development of several variations of the strategy,
such as scaffold-building to “automatically” produce a sequence map as a shotgun
project proceeds. This was described and simulated by Roach and colleagues
(Roach et al., 1995). Also variations of the approach were detailed by other groups
(Chen et al., 1993; Smith et al., 1994; Kupfer et al., 1995; Nurminsky & Hartl,
1996). TIGR Assembler (Sutton et al., 1995) was the first assembly program to
utilize the clone length information, and was successfully applied to assemble the
Haemophilus influenzae genome by the whole genome shotgun method. Myers
engineered an approach to the assembly of larger genomes, using paired wholegenome shotgun reads (Anson & Myers, 1999). Myers’ approach was implemented
in the Celera Assembler and used to succesfully assemble first the Drosophila
melanogaster genome (Myers et al., 2000). The algorithm has also been applied
to the human genome.
On the realization that the initial, correct determination of overlaps was critical
to the overall assembly result, the reads were first conservatively clipped to avoid
the lower accuracy regions at the ends, where the overlaps are typically detected.
This was first implemented in 1991 (Dear & Staden, 1991) and in 1995, in the
GAP (Bonfield et al., 1995) programs , followed by many others. The base quality
estimates were first used to improve the consensus calculation (Bonfield & Staden,
1995). The major improvement in assembly followed from the base calling program
Phred (Ewing et al., 1998; Ewing & Green, 1998), that computes error probabilities
for base calls using parameters derived from electropherograms. The idea of error
10
probabilities was earlier described by several groups (Churchill & Waterman, 1992;
Giddings et al., 1993; Lawrence & Solovyev, 1994; Lipshutz et al., 1994). Other
base calling programs have followed, that compute error probabilities for base calls,
and in addition, gap probabilities, for example Life Trace (Walther et al., 2001).
The Phred error probabilities were utilized in Phrap (http://www.phrap.org) to
discriminate pairwise overlaps.
A different approach was used in the design of GigAssembler (Kent & Haussler,
2001). The GigAssembler was designed to work as a second-stage assembler and
order initial contigs that were assembled separately by Phrap or another assembly
method. The program utilizes several sources of information, e.g. paired plasmid
ends, expressed sequence tags (ESTs), bacterial artificial chromosome (BAC) end
pairs and fingerprint contigs. The GigAssembler algorithm was used to assemble
the public working draft of the human genome.
2
2.0.1
Milestones in Genome Sequencing
First Cellular Organisms
The first bacterial genome, Haemophilus influenzae Rd 1.83 Mbp (Fleischmann
et al., 1995), sequenced using the whole genome shotgun sequencing approach.
First eukaryotic organism Saccharomyces cerevisiae 13 Mbp (Goffeau et al., 1996).
A bacterium, Escherichia coli 4.60 Mbp (Blattner et al., 1997) is a preferred model
in genetics, molecular biology, and biotechnology. E. coli K-12 was the earliest
organism to be suggested as a candidate for whole genome sequencing.
2.0.2
First Multicellular Organisms
A small invertebrate, the nematode or roundworm, Caenorhabditis elegans 97 Mbp
(C. Elegans Sequencing Consortium, 1998).
A fruit fly, Drosophila melanogaster 137 Mb (Adams et al., 2000).
Homo sapiens 2.9 Bbp International Human Genome Sequencing Consortium,
Nature, Feb. 15, 2001 and Venter et al. Science Feb. 16, 2001.
Mus Musculus 2.6 Bbp Celera Genomics, spring 2001, unpublished.
According to Genomes Online Database (Bernal et al., 2001), on 10th of April,
2002, the total of 84 complete genomes have been published, 271 on-going sequencing projects of prokaryote genomes and 178 eukaryotic sequencing projects.
The development of computer hardware and powerful computational tools is central to large-scale genome sequencing. Without fragment assembly software and
automation of gel reading by image-processing tools it would not have been feasible
to assemble all gel reads up to millions of bases of contiguous sequence.
11
3
Shotgun Fragment Assembly Algorithms
The goal of a shotgun fragment assembly program is to order overlapping reads
to recreate the original DNA sequence. In Practice this means the construction
of the “best guess” of the original clone sequence, BAC, Cosmid or in the case of
whole genome shotgun, genome sequence. In order to reach the goal of revealing
the original DNA sequence, several problems and complications must be solved.
Many real world problems are computationally intractable. The assembly of shotgun sequences contain such problems in multiple alignment and fragment layout
determination stages. Hence, in many cases it will not be possible or feasible to
compute the optimal solution and therefore there is no quarantee that the resulting
consensus sequence is the same as the original DNA sequence.
3.1
Types of Problems in Sequence Assembly
An algorithm is a procedure that specifies a set of well-defined instructions for
a solution of a problem in a finite number of steps. Apart from the fact that it
is difficult to prove that an algorithm is correct, even when it solves what it is
intended to solve, the central question in computer science is the efficiency of an
algorithm: How the running time, and number of steps needed to get a solution
depends on the size of input data. This depends mainly on the problem type at
hand. The problems can be divided into two main complexity classes: polynomial
and non-polynomial. In general polynomial time problems are efficiently solvable,
whereas non-polynomial problems are hard to solve since the running time of an
algorithm grows faster than any power of the number of items required to specify
an increasing amount of input data. No practical algorithm that solves a nonpolynomial problem exists, since it would take an unreasonably long time to get
an answer (Garey & Johnson, 1979).
It is very hard to prove that a problem is non-polynomial, since all possible algorithms would have to be analyzed and proved not to be able to solve the problem
in polynomial time. Several problems exist that are believed to be non-polynomial,
but this has not yet been proven. For example, the well known Travelling Salesman
Problem (Karp, 1972): Given a cost of travel between cities, the problem is to find
the cheapest tour that passes through all of the cities in the salesman’s itinerary
and returns to the point of departure. Finding prime factors of an integer is also
believed to be non-polynomial and many security systems that encrypt e.g. credit
card numbers rely on this belief.
In 1971, Steve Cook (Cook, 1971) introduced classes of decision problems, polynomial runnning time P , and nondeterministic polynomial running time N P 2 ,
based on the verification of the solution. P is the class of decision problems for
which the solution can be both verified and computed in polynomial time. For the
N P -class of problems it is possible only to verify a given answer in polynomial
time. A problem is in the N P -complete class, if it is possible to solve all problems
2N P
is not the same as non-P .
12
within the class in polynomial time, using an algorithm that can solve any one
of the problems. In other words, any problem within a class can be converted to
any other problem within the same class. N P -hard problems are a large set of
problems that include N P -complete problems so that an algorithm that can solve
an N P -hard problem, can solve any problem in N P with only polynomially more
work. Since Cook’s theorem, the travelling salesman problem has been proven to
be N P -complete, so have a large number of other problems.
Many real world problems are inherently computationally intractable. This fact
led to the development of the theory of approximation algorithms dealing with
optimization problems. The objective of optimization is to search for feasible solutions by minimizing or maximizing a certain objective function when the search
space is too large to search thoroughly and thus when computing the optimal solution would take too long. Different N P optimization problems have different
characteristics. Some problems may not be approximated at all. Other optimization problems differ in regard to how close to the optimum an approximation can
get.
Based on empirical evidence, it has been conjectured that all N P -complete problems have regions that are separated by a phase transition and that the problems
that are hardest to solve occur near this phase transition. This seems to be due to
the large number of local minima between the two regions that are either under
constrained or over constrained (Cheeseman et al., 1991).
3.2
Complications in Fragment Assembly
There are a number of computational problems that a sequence assembly program
has to be able to handle. The most important ones are:
1. Repeated regions. Repeats are difficult to separate and often cause the fragment assembly program to assemble reads that come from different locations.
Short repeats are easier to deal with than repeats that are longer than an
average read length, of v 500 bp. Repeats may be identical or contain any
number of differences between repeat copies. There are interspersed repeats
which are present at various different genomic locations, and tandem repeats, a sequence of varying length repeated one or several times at the same
contiguous genomic location. Repeats present the most difficult problem
in sequence assembly. It may be close to impossible to correctly separate
identical repeats, but fortunately identical repeats are rare since different
copies have accumulated mutations during evolution. This can be used to
discriminate different repeat copies (Figure 1.).
2. Base-calling errors or sequencing errors. The limitation in current sequencing technology results in varying quality of the sequence data between reads
and within each read. In general the quality is lower toward the ends, where
the overlapping segment between a read pair occurs. There are a number of
reasons for this. The data at the end of the reads originates from the longest
fragments and the proportional difference between long DNA fragments, dif-
13
Genome sequence:
Repeat
Repeat
CTTCGCGTCATCATCACTTGAGTCATCATCACCTCGGA
Sequence reads in the correct layout:
CTTCGCGTCATCATCA
TCATCATCACTTGA
CTTGAGTCATCATCA
TCATCATCACCTCGGA
Due to repeats, there is an alternative, incorrect layouts:
Contig 1:
Contig 2:
CTTCGCGTCATCATCA
TCATCATCACTTGA
TCATCATCACCTCGGA
CTTGAGTCATCATCA
1. One contig including some erroneous sequencing errors:
CTTCGCGTCATCATCA
TCATCATCAC*TTG* A
CTT*GAGTCATCATCA
TCATCATCACCTCGGA
Figure 1: A fictious genome sequence contains a repeat which is present is two
copies.
fering only one base, is smaller than of that compared to short fragments.
Thus, the random incorporation of ddNTPs results in a geometric distribution of sizes, yielding a weaker fluorescent signal for the large molecules.
Molecules diffuse in the gel and the longer the time takes for fragments to
pass the laser detector, the more they diffuse. These factors among others
effectively prevent considerably longer reads than approximately one kilobase. In routine production the read length varies from about 400 bases
to 900 bases. The sequence reads may also have low quality regions in the
middle of high quality regions. These can be caused by for example simple
repeats and “compressions”, i.e. some bands in the sequencing gel run closer
together than they are expected to, caused by secondary structure effects.
Single stranded DNA can self anneal to form a hairpin loop structure which
causes it to move faster on the gel than would be expected according to its
length. This is a fairly common phenomenon in GC-rich segments and results
14
in compression of trace peaks. There are four different types of sequencing
errors: 1. Substitutions: A base is erroneously reported. 2. Deletions: One
or several bases are missing. 3. Insertions: One or several erroneous extra
bases are reported. 4. Ambiguous base calls: A,T,G, or C. See Table 1.
Symbol
G
T
U
Y
K
W
B
D
Meaning
G
T or U
U or T
T, U or C
G, T or U
A, T or U
G, T, U or C
G, A, T or U
Symbol
A
C
R
M
S
H
V
N
Meaning
A
C
G or A
A or C
G or C
A, C, T or U
G, C or A
G, A, T, U or C
Table 1: IUPAC ambiguity codes
3. Contamination. Sequences from the host organism used to clone shotgun
fragments, e.g. E.coli, which are mixed with shotgun sequences.
4. Unremoved vector sequences present in reads. If not removed, these may
cause false overlaps between reads.
5. Unknown orientation. It is not known from which strand each fragment
originates. This increases the complexity of the assembly task. Hence, a read
may represent one strand or the reverse complement sequence on the other
strand. A BAC-sized project may contain over 2000 reads and since it is not
known from which strand each read originates, a duplicate, complementary
set of reads is created by first reversing the base sequence and complementing
each base. This results in a total of over 4000 reads (Figure 2.).
Figure 2: The orientation of each fragment is initially unknown.
6. Polymorphisms. Complicates separation of nearly identical repeats. Polymorphisms will not usually be detected through “clone by clone” sequencing
since only one variant for each genomic region is sampled.
15
7. Incomplete coverage. Coverage varies in different target sequence locations
due to the nature of random sampling. The coverage has theoretically a
certain probability to be zero depending on the average sampling coverage of
the target genome, see equation 1. The number of gaps are further increased
by bias due to the composition of the genomic sequence, e.g. some sequences
may be toxic and interfere with the biology of the host organism.
3.3
Base calling
Commercial companies, such as Applied Biosystems that among other things produces DNA sequencers, have not made their algorithms public. However, public
research for base calling on fluorescent data has been performed. Phred (Ewing
et al., 1998) is today a popular base calling system that has challenged the ABI
base calling. Tibbet, Bowling, and Golden(Adams et al., 1994) have studied base
calling using neural networks and Giddings et al. (Giddings et al., 1993) describe
a system that uses an object-oriented filtering system. Also, Dirk Walther et al.
(Walther et al., 2001) describe a base calling algorithm reporting lower error rates
than Phred.
Base-calling refers to the conversion of raw chromatogram data into the actual sequence of bases figure 3. Using the Sanger sequencing method, extension reactions
are performed using cloned or PCR product DNA segments as templates. The extensions generate products of all lengths as determined by the sequence. When
these products are run on an electrophoresis gel, the base sequence can be read
starting from the bottom. Different bases can be identified by labelling each base
reaction with a different fluorescent dye. In sequencing machines using slab-gel or
capillary technique, the dyes are exited with laser and the emitted light intensity
is usually detected by a CCD-camera or a photo multiplier tube. This produces a
set of four arrays called traces corresponding to signal intensities of each color over
a time line. These traces are further processed by mobility correction, locating
start and stop positions, base line substraction, spectral separation, and resolution
enhancement. Base calling refers to the process of translating the processed data
into an actual nucleotide sequence.
Figure 3: A graphical view of the four traces, A, T, G and C showing the Phred
and ABI base-calling.
16
3.4
The Data Quality
The limitation in current sequencing technology results in varying quality of the
sequencing data between reads and within each read. In general the quality is
lower towards the ends. There are number of reasons for this. The data at the
end of the reads originates from the longest fragments and the proportional difference between long DNA fragments, differing only one base, is smaller than of
that compared to short fragments. Also the random incorporation of ddNTPs
results in a geometric distribution of concentration and size yielding a weaker fluorescent signal. Then there is the time factor. Molecules diffuse in the gel and
the longer the time until the fragments pass the laser detector more they diffuse.
These factors among others effectively prevent considerably longer reads than approximately one kilobase. In routine production the read length varies from about
400 bases to 800 bases. The sequence reads may also have low quality regions
in the middle of high quality regions. These are mainly caused by the characteristics of polymerase and sequencing reactions. The base sequence of DNA itself
has influence on the data quality. Single stranded DNA can self anneal to form a
hair pin loop structure which causes it to move faster on the gel than would be
expected according to its length. This is a fairly common phenomena on GC-rich
segments and results in compression of trace peaks. Figure 4. Considering the
Figure 4: An example of compressed traces.
above it is not suprising that there have been a lot of interest in quality assesments in DNA sequencing during the last decade (Ewing & Green 1998(Ewing &
Green, 1998), Richterich 1998(P.Richerich, 1998), Li et al. 1997(Li et al., 1997),
Bonfield & Staden 1995(Bonfield et al., 1995), Naeve et al. 1995(Naeve et al.,
1995), Lawrence & Solovyev 1994(Lawrence & Solovyev, 1994), Lipshutz et al.
1994(Lipshutz et al., 1994), Khurshid & Beck 1993(Khurshid & Becks, 1993) and
Chen & Hunkapiller 1992(Chen & Hunkapiller, 1992))
3.5
Error Probabilities
There are several obvious reason for why we would want to have error free sequence
data. The fragment assembly problem would be algorithmically much simpler and
17
the significantly lower cost associated with sequencing since fewer reads would
be needed. Unfortunately, we cannot get perfect data, but we have to settle
with the second best and try to measure the accuracy. In fact, many early base
calling algorithms estimated confidence values (Giddings et al., 1993; Giddings
et al., 1998; Golden et al., 1993) which is best done by studying electropherogram
data. In 1994 Lawrence and Solovyev (Lawrence & Solovyev, 1994) carried out
an extensive study defining a large number of trace parameters and an analysis
to determine which ones are most effective in distinguishing accurate base calls.
After 1998 when Brent Ewing and Phil Green implemented Phred (Ewing & Green,
1998), a base-calling program with the ability to estimate a probability of a basecall, Phred became one the most widely used base-calling programs today and Pred
quality values became a standard within the sequencing community (P.Richerich,
1998). These quality values are logarithmic and range from 0 to 99 where increased
value indicates increased quality. They are defined as follows:
q = −10 · lg P
Where q is the quality and P is the estimated error probability of a base-call.
Phred quality values have a wide range. For example, a quality value 10 means
1
that the probability of a base being wrong is 10
or having a 90% accuracy. An
99.99% accuracy is required for finished shotgun data submitted into data bases,
which means that the probability of an error is 10 1000 .
Now we know that the root of many of the problems in fragment assembly essentially lies in the error prone base-calling. Ideally peaks in all four traces, one
for each base, would be Gaussian-shaped with even distances without any background. As always in real life, several factors disturb the obtainable data quality,
resulting in variable peak spacing, uneven peak heights and variable background.
Thus, resulting in erroneous final nucleotide sequence. One of the key essences
in successfull definition predictive error probabilities is the correct characterization of traces. Lawrence and Solovyev (Lawrence & Solovyev, 1994) considered
single peaks in traces, but according to the work by Ewing and Green (Ewing &
Green, 1998), the measures become more reliable when peaks in the neighborhood
are taken into account. More information is available since indications of error
may be present in near-by peaks but not in the peak under consideration. Ewing
and Green found the following parameters to be the most important to effectively
detect errors in base-calling:
1. Peak spacing, D. The ratio of the largest peak-to-peak distance, Lmax to the
max
smallest peak-to-peak distance, Lmin , D = L
Lmin . Computed in a window of
seven peaks which is centered on the current peak. This value measures the
deviation from the ideal when peaks are evenly spaced, yielding a minimum
value of one.
2. Uncalled/called ratio in window of seven peaks. The ratio of the amplitude of
the largest peak not resulting in a base call to the amplitude of the smallest
peak resulting in a base call, i.e. called peak. If the current window lacks
an uncalled peak, the largest of the three remaining trace array values at
the location of the called peak is used instead. In case the called base is an
’N’, phred vill assign a value of 100.0. The minimum value is 0 for traces
18
with no uncalled peaks. With good data one of the four fluorscent signals
will clearly dominate over the three other signals. Computed in a window of
seven peaks centered at the base of interest.
3. Uncalled/called ratio in window of three peaks. The same as above, but using
a window of three peaks.
4. Peak resolution. The number of bases between the current base and the
nearest unresolved base. Base calls tend to be more accurate if the peaks
are well separated and thus do not overlap significantly.
By re-sequencing an already known DNA sequence and using it as a golden standard, it is possible to compare the output of a base-caller program to this standard.
The errors can now be categorized using the parameter values computed from the
traces for each base. This can be used in predicting errors in unknown sequece
by comparing the parameter values of the known errors frequencies. In order to
compute new errors probabilities rapidly for unknown sequences, a lookup table is
created where all four parameters values for each category of error rate is stored.
Phred uses a lookup table consisting of 2011 lines to predict the error probability
and to set the quality values.
The Phred quality values provide only information on the probability of a base
being errorneously sequenced. Recall that there are four different types of errors
that may occur in base-calling: substitutions, deletions, insertions and ambigous
calls. Phred quality values do not give us a probability of a missing base, i.e.
a probability of deletion. A recently published base-calling program, Life Trace
by Dirk Walther et al. (Walther et al., 2001) introduces a new quality value,
gap-quality. Pre-compiled versions of Life Trace for several different platforms are
available at http://www.incyte.com/software/home.jsp.
The error probabilities can be used in several ways in fragment assembly programs.
For example screening and trimming of reads and in defining pair-wise overlaps of
reads. We discuss this more throughly in part 3.9
3.6
Genome Coverage
The benefit of the shotgun method is that several reads usually cover the same
region on both DNA strands and the a sought after consensus sequence can be
computed using the information from several reads. Usually in a genomic sequencing project the coverage is about 8, often denoted as 8X. Coverage is defined as
follows:
N ·r
C=
(2)
G
Where c = Coverage, N = Number of reads, r = Mean read length and G =
Length of the genome. How many reads do we need to get 10X coverage of a BAC
sequence of length 125 kb, if the mean read length is 480 bases?
N=
c·G
r
000 = 2604 reads
= 10·125
480
19
About 2600 fragments need to be sequenced to get 10X coverage. Remember
that we don’t know the orientation of any of the fragments, therefore we need to
complement all reads, i.e., compute the reverse complement sequence and then
figure out the orientation in fragment assembly. This yields 2 · 2600 = 5200 reads,
that an assembly program has to combat.
Figure 5: A assembled with 10X average coverage. y-axis shows the coverage and
x axis the position on the .
This is the total average coverage, but the actual observed coverage will vary along
and between the assembled . The observed coverage is zero on regions that are
not sequenced at all. Figure 5. shows a typical coverage variation along a which is
sequenced using 10X average coverage.It may be valuable to know the probability
to observe a coverage of certain depth or a resulting mean number of gaps and
their average length. If we know the mean read length, r, average coverage, c and
the length of the genomic region, G, the coverage can be computed using C as
a binomial random variable. The shotgun sequencing can be seen as randomly
selecting an initial base as a starting position within the clone we aim to sequence
and then continuing from that position forward an average read length number
of bases. What is the probability of selecting this initial base? Let the average
coverage be c = 8, the genome length G = 104 bases and the average read length
r = 500 bases. Using the equation (2), we can calculate that we have to sequence
1600 fragments to achieve the average coverage of 8. Thus, we will select a starting
base for sequencing 1600 times, which gives us the probability of randomly selecting
a base position,
No. of reads
1600
p=
=
= 0.016.
G
104
Each selection or ’trial’ of a starting base results in ’success’ with the probability,
p = 0.016 and in ’failure’, not selected with the probability 1 − p = 1 − 0.016 =
0.984. How many trials do we make? This may be a little tricky. The number
of ’trials’ is the same as the average read length. We succeed to get one unit of
coverage by selecting one of the positions in a window that equals the average read
lengt. The probability of a succesful selection
is p and the number of different ways
¡¢
we can select a base in the window is rc . If we think coverage C as the number
of “successes”, the probability to observe a certain coverage, C, can be computed
using the binomial distribution function:
P {C ≤ i} =
i µ ¶
X
r
k=0
k
pk (1 − p)r−k ,
20
i = 0, 1, ..., r
(3)
Where p is the probability of choosing c coverage times a base from the number
of bases equaling the mean read length, r:
p=
c
r
(4)
Note that we can derive the equation (4) starting from equation (2). C is binomial
with parameters (r,p).
Example 1. Let coverage c = 8, the mean read length, r = 500 and the length
of the clone, G = 104 . The probability of observing zero coverage, C is:
µ
¶500
8
P {C = 0} = 1 · 1 · 1 −
≈ 3.14 · 10−4 ,
500
and the probability of observing a coverage, C = 12 is
P {C = 12} =
µ ¶µ
¶12 µ
¶500−12
500
8
8
1−
≈ 4.79 · 10−2 ,
12
500
500
and the probability of a coverage, C ≤ 1 is
P {C ≤ 1} = P {C = 0} + P {C = 1}
µ
3.14 · 10−4 +
500
1
¶µ
8
500
¶1 µ
¶500−1
8
1−
≈ 2.56 · 10−3
500
The Poisson random variable can be used to approximate the binomial random
variable with parameters (n,p) when p is small and n large so that np is of moderate
size. λ = np
The Poisson probability mass function:
p(i) = P {X = i} = e−λ ·
λi
i!
(5)
The probability that a base is not sequenced: P0 = e−C , where C is the coverage.
The total gap length = L · e−C , average gap size = L
n , where n is the number
of randomly sequenced segments (Lander & Waterman, 1988; Fleischmann et al.,
1995).
3.6.1
Exercises
1. Average coverage is 3.5, the mean read length 420 bases and the length of
the BAC clone is 120 kb. Calculate the probability of a base not being
sequenced.
21
2. Suppose you have a BAC sequence that contains two repeated regions of
length 1 kb. The repeat copies differ 0.8% from each other and the mutations
are evenly distibuted. You want that at least two differences are covered by
at least four same reads with a porbability of 99.9%. What average coverage
should you have?
3.7
Distance Constraints
When clones of DNA are sequenced from both ends, the approximate distance
and their relative orientation is known. These sequence pairs from cloned ends
are called mate-pairs (Myers et al., 2000) or sequence mapped gaps (Edwards
& Caskey, 1990). The distance information can be used to both verify finished
assemblies and as an additional source of information in assembly algorithms. The
modified shotgun sequencing strategy using these distance constraints is coined
double-barreled. Typically a whole-genome sequencing projects use a mixture of
clones containing inserts of several lengths, e.g. 2 kbp, 5 kbp, 50 kbp and 150 kbp
clones.*FIGURE* A usual way to employ the information in fragment assembly
programs, is to build a so called scaffold. A scaffold is a set of or reads that
are positioned, oriented and ordered corresponding to the distance information
provided by cloned ends. *SCAFFOLD FIG* Only an approximate length of
clones are known and it is therefore not possible to rely only on one pair of cloned
ends to build an assembly scaffold. To acquire a reasonable confidence, a certain
coverage of mate-pairs is required. This is obviously dependent on the accuracy
of the laboratory procedure when fragmented DNA seqments of a specific size
are chosen. For example in the whole-genome shotgun assembly of Drosophila, a
single mate-pair information was found to be wrong 0.34% of the time, while two
mate-pairs providing consistent information, the error was found to decrease to
10−15 (Myers et al., 2000).
3.8
Fragment Assembly Problem Models
The sequence reconstruction or fragment assembly problem would be easy if the
original DNA sequence was unique, i.e. did not contain repeats. If we find a common sub-sequence between two different reads 14 bases of length, the probability
1
of finding an identical sequence by random is 4−14 = 268 435
456 . This means
that if the length of the sequence we are assembling is about 270 Mbps, we would
expect to find one such sequence by cheer chance. In a BAC-sized project ( 100000
−4
bps) the probability of a random match is 268100000
435 456 ≈ 4 ∗ 10 , provided that
the DNA sequence is random3 . To solve the fragment assembly problem would be
easy, since we would only need to find all the matching sub-sequences between all
the reads and with a high probability almost all the overlaps assigned this way,
would be correct. All what is left to do is to assemble these reads into contigous
sequences, contigs, compute consensus sequence and we are done. Unfortunately,
3 NOTE that this is NOT the probability of two reads sharing a common sequence of a certain
length.
22
genomes usually are not random and contain repeats. This tactics is doomed to
fail in practice.
During decades, many algorithms have been developed that attempt to solve the
fragment assembly problem. One of the keys to good fragment assembly is a
successful definition of overlaps between reads. These overlaps can be of four
different types, figure 6. Read A can overlap read B either at the
Figure 6: True overlaps between reads can be of four different types. In all cases
the overlap must end at least at the end of one of the reads.
beginning or at the end, read A can be contained by read B or vice verse. In
all cases the overlap has to end at the end of at least one of the reads to be
considered as a true candidate overlap. All true overlaps fall into one of these
categories. Considering that we are interested in pairing reads that come from the
same clone, i.e. genomic region and should therefore have identical base sequences,
it is surprisingly hard to define these overlaps correctly. The reads contain many
sequencing errors. In general the most of the errors are at the beginning and at
the end of the reads, making it hard to define where exactly the read begins and
where it ends, figure 7.
Figure 7: A typical quality distribution of a shotgun read. The beginning and the
end of a read has low quality. X-axis: The position (bases) on the read, Y-axis:
The PHRED quality value.
All fragments usually pass through a trimming process where low quality end are
trimmed. If trimming is too stringent, information is lost and a greater number
of reads is required to give a certain coverage. See section 3.6. On the other
23
hand, if the trimming is too loose leaving many sequencing errors, it is difficult
to determine whether the sequences come from the same region or if they just are
very similar. Since many of the defined overlaps between reads are not right, it
is very hard to rebuild the original genomic sequence. There will be extremely
many ways to put this puzzle together. Actually so many ways, it is impossible
to try out all the possibilities. It gives rise to a combinatorial explosion. This
type of problem is very hard to solve and in computer science it is called nondeterministic polynomial complete, NP-complete. This is the hardest problem in
NP. Thousands of problems have been proven to be NP-complete. One of the most
famous is the Traveling Salesman Problem. Many computer scientists are working
to find general solutions for this problem type. The simplest algorithm to solve
this problem is the greedy algorithm.
3.8.1
The Shortest Common Superstring Model
The fragment assembly problem can be abstracted into the shortest common superstring problem (SCS). The object is to find the shortest possible superstring
S that can be constructed from a given set of strings F so that S contains each
string in F as a substring. SCS can also be be reduced to the traveling salesman
problem and also has applications in data compression.
Maier and Storer showed that SCS is N P -complete (Maier & Storer, 1977). Later,
in 1980, a more elegant proof was described (Gallant et al., 1980). Hence SCS is
N P -complete, no feasible polynomial algorithm is known that solves this problem,
but polynomial approximation algorithms exist. Moreover, SCS was proven to be
M AX − SN P -hard in 1994 (Blum et al., 1994), meaning that no polynomial-time
algorithm exists that can approximate the optimal solution within an arbitrary
constant, but rather within a predetermined constant (section 3.1). A simple
greedy algorithm can be used to approximate the solution by repeatedly choosing
a maximum overlap between substrings in the set F until only one string is left.
The greedy algorithm was first used in fragment assembly in 1979 (Staden, 1979).
Tarhio & Ukkonen and Turner established some performance guarantees for the
greedy algorithm, which implied that the resulting superstring S is at most half the
total, catenated length of the substrings in F (Tarhio & Ukkonen, 1988; Turner,
1989). Blum et al. were the first to describe an approximation algorithm for SCS
with an upper bound of three times optimal (Blum et al., 1991). The approximation methods have been continuously refined. Kosaraju et al. established the
3
bound of 2 50
63 (Kosaraju et al., 1994), Stein and Armen the bound of 2 4 in 1994
2
and 2 3 bound in 1996 (Armen & Stein, 1994; Armen & Stein, 1996) and Sweedyk
has reduced the bound to 2 12 (Sweedyk, 1995). It has been conjectured that greedy
is at most two times optimal.
SCS is a sub-optimal model for fragment assembly, mainly because it tends to
collapse repeated regions and it accounts for neither sequencing errors (i.e. substitutions, insertions and deletions) nor unknown orientation.
24
3.8.2
The Sequence Reconstruction Model
The problem model derived from SCS that accounts for sequencing errors and
for fragment orientation is called The Sequence Reconstruction Problem (SRP).
The minimum number of edit operations, i.e. substitutions, insertions, deletions
required to convert the sequence a into the sequence b is defined as the edit distance
between sequences and denoted d(a, b). Given a set of sequence fragments, F and
an sequencing error rate ² ∈ [0, 1], the object is to find the shortest sequence S
such that for all fi ∈ F there is a substring Ssub of S such that
min{d(Ssub , fi ), d(Ssub , fir )} ≤ ²|Ssub |,
where fir is the reverse complement of fi .
Since the sequence reconstruction problem can be reduced to N P -complete SCS
with zero error rate, SRP is also N P -complete. The proof was given in 1996
(Waterman, 1996).
Peltola et al. (Peltola et al., 1984) were the first to include errors and fragment
orientation in the problem model together with graph theory, and they used the
greedy algorithm to approximate the fragment layout. Also, Huang included the
error criterion in the CAP assembly program (Huang, 1992). Kececioglu and Myers describe the extension to SRP by introducing discrimination of false overlaps
based on error likelihood including an extensive treatment of the unknown orientation problem using interval graphs (Kececioglu, 1991; Kececioglu & Myers,
1995). Phrap uses Phred quality values in a likelihood model to discriminate false
overlaps (section 3.9.2 and 3.9.3).
The sequence reconstruction model complies better with the reality than the shortest common superstring model.
3.8.3
Graphs
An article by L. Euler started a branch of mathematics called graph theory (Euler,
1736; N. Biggs & Wilson, 1976). A graph is a collection of vertices connected by
edges. Since each read should form an interval on the target sequence, it is natural
to model this problem as an interval graph problem, with reads corresponding to
vertices and overlaps to edges. The directed graph Gov (V, E) in figure 8 for the
sequence S = GT GAT GCT GG has a vertex set V = GTG, TGA, GAT, ATG,
TGC, GCT, CTG, TGG. Each edge E is an overlap with an adjacent sequence
by the minimum of two bases. A Hamiltonian path in the graph is a path that
passes through every vertex exactly once. However, a Hamiltonian path does not
necessarily pass through all the edges. A Hamiltonian cycle or circuit is a Hamiltonian path that ends in the same vertex as that in which it started. Each edge
in the graph can be weighted by a score that reflects the quality of an overlap and
the traveling salesman problem formulation can be used to find the Hamiltonian
path that yields the best score. This problem is however N P -complete. If the
amount of input data is large, and in particular in the presence of similar repeats
25
Figure 8: Directed graph Gov for the sequence GTGATGCTGG with the vertex
set V . There is one Hamiltonian path that visits all vertices.
and sequencing errors, it might be impossible to find the optimal solution within
a reasonable time.
An alternate data structure can be used to find Euler paths. An Euler path in a
graph is a path that travels along every edge exactly once, but might pass through
individual vertices of the graph more than once. An Euler path that begins and
ends in the same vertex is an Euler circuit or tour. The redundant shotgun data
can be converted into an Euler path problem by producing k-tuple data. This
is done by extracting all overlapping k-tuples from all shotgun reads and then
collapsing all edges to single edges where data is consistent so that only unique
k-tuples remain. The graph is constructed on (k − 1)-tuples so that each edge
has a length of one. For each read fragment f of length k in S an edge from the
vertex labeled with the left-most k − 1 bases of f is directed to the node labeled
with the right-most k − 1 bases of f . Edges may be weighted by e.g. the number
of fragments associated with an edge. An example of a directed graph Geu is the
case shown in figure 9, that is constructed for the sequence S with k = 3.
The fragment assembly problem is now recast into the problem of finding an Euler
path in the graph and there are efficient algorithms for finding such paths. The
reduction of sequencing by hybridization (SHB) to a question of finding Euler paths
was originally developed by Pevzner (Pevzner, 1989) and the hybrid algorithm,
based on SHB, was described by Idury and Waterman (Idury & Waterman, 1995).
This problem is not N P -complete. However, this gain is paid partly by some loss
of information in the process of fragmenting the original input data into a number
of k-tuples. There may be several Euler paths in one graph, as shown in figure 9,
that start at the vertex labeled GT and results in the sequences shown in figure
10.
One approach of getting the set of all Euler paths in a graph is based on the
fact that all spanning trees can be enumerated and that there is a relationship
between the set of all spanning trees and the set of all Euler paths in a graph
(Even, 1979). Another approach is to transform any compatible string by a series
26
Figure 9: The graph Geu for the sequence S = GT GAT GCT GG
GTG
TGA
GAT
ATG
TGC
GCT
CTG
TGG
---------1. GTGATGCTGG
GTG
TGC
GCT
CTG
TGA
GAT
ATG
TGG
---------2. GTGCTGATGG
Figure 10: Euler path solutions for the graph Geu .
of simple string operations, so that each string produced is itself compatible with
the input data. This is based on a conjecture described by Ukkonen and proved
by Pevzner (Ukkonen, 1992; Pevzner, 1994).
The Euler path method produces single sequences as output. A multiple alignment
is required to evaluate the underlying read data. This can be accomplished by
aligning all the initial reads against the Euler sequence(s).
Idealized versions of problem models are rarely applicable in practice. Sequencing
errors, in particular together with nearly identical repeats, are a complicating
factor. Overlaps are based on exact sequence identity, thus there is a possibility of
losing existing overlaps because the identities are distorted by sequencing errors.
The input data may be trimmed to be of very high quality, yet this results in
shorter read lengths. However, it is possible to try to correct the errors prior to
the construction of the graph, as is the approach used in the EULER assembler
(Pevzner et al., 2001).
27
3.9
Algorithms
The fragment reconstruction problem based on overlaps can in turn be divided
into a number of sub-problems. 1. Discarding contamination, i.e. sequences from
vector and chimeric reads and the host organism. 2. Determination of all overlaps
between reads. 3. Computation of the layout of the contigs, i.e. the orientation
and relative position of the reads in contigs. 4. Creation of multiple alignments
of all reads. 5. Computation of the consensus sequences.
3.9.1
Contamination
Contaminating reads can easily be discarded by screening the reads against a
database consisting of the host genome sequences, as well as the vector sequences.
Since chimeric reads are composites, consisting of at least two sequences originating
from different locations of the target genome, they can be recognized by their
pattern of overlap with other reads. One such algorithm was described in 1996
and implemented in CAP2 (Huang, 1996).
3.9.2
Overlap Detection.
In order to determine all pairwise overlaps, an all-versus-all comparison of reads has
to be performed. For example, 2000 reads would require (2 · 2000)2 = 16,000,000
pair-wise comparisons, since each possible orientation is tested. The classical solution to this problem of approximate string matching is a dynamic programming
method (Sellers, 1980). The disadvantage of this approach is the quadratic running
time, O(nm), where n and m are the lengths of the reads. There are several approaches that improve the running time by only computing a part of the dynamic
programming matrix, for example Ukkonen who reduces the time to O(nk), where
k is the edit distance (Ukkonen, 1985). The performance can be further improved
by pre-prossessing the database using indexing techniques, such as patricia trees
(Knuth, 1973), suffix trees (McCreight, 1976) and suffix arrays (Manber et al.,
1990). These structures allow a query time proportional to the length of the query
string, while being independent of the size of the database, which can be constructed in linear time. The time to construct the database can be amortized by
performing a large number of queries.
There are many different approaches to solving the all-versus-all comparison problem (Gusfield, 1997; Chen & Skiena, 1997). One solution that was used in the
TIGR Assembler, was to locate all k-tuples shared between read pairs, similar to
the initial processing in BLAST (Altschul et al., 1990). A more practical solution,
that can be applied to shotgun reads, is to use a combination of suffix arrays and
indexes built of the database sequences (Myers, 1995a). A suffix array can be
constructed by indexing all instances of short words, typically 10 to 14 bases, that
occur in the database. All instances of words are stored in an array. By assigning
pointers to point to their locations in reads, the lists of suffixes in each array position can be sorted in lexicographical order. In this way, all similar reads will be
28
grouped close to each other and it is possible to rapidly extract matching suffixes.
The candidate alignments are created by dynamic programming, where only a
band in the dynamic programming matrix is computed (Chao et al., 1992), or a
variation of the Smith-Waterman algorithm (SWA) (Smith & Waterman, 1981) is
used. This method is used in the Phrap fragment assembly program and similar
methods with some variation are used in many other assembly programs, e.g. in
FALCON (Gryan, 1994) and ARACHNE (Batzoglou et al., 2002). Approaches
based on hybridization and techniques based on the sampling of large numbers of
segments, sub-words (Kim & Segre, 1999), have not yet been shown to be successful
in practice, mainly due to the presence of ubiquitous repeats. Each of the pair-wise
alignments are scored in order to evaluate the quality of overlaps. The total score
is obtained by dynamic programming using a scoring scheme with specified match
score, and using mismatch and gap penalties (Pearson & Lipman, 1988; Altschul
et al., 1990), possibly together with some estimate of the total sequencing error, as
in the assembly programs, CAP and CAP2 (Huang, 1992; Huang, 1996). A more
accurate approach is to utilize quality values, i.e. the probability that a base is
erroneously called. Most fragment assembly programs that use error probabilities,
use quality values generated by Phred, which is the most widely used base calling
program. Several programs use the error probabilities merely as weights for match
score, mismatch and gap penalties, e.g. STROLL (Chen & Skiena, 2000) and CAP
(Huang & Madan, 1999). Phrap uses Phred quality values in a likelihood model
to discriminate false overlaps. The initial candidate overlaps are initially filtered
by a minimum SWA score and then all alignments are re-scored by computing a
log likelihood ratio (LLR), which is used in the construction of contigs.
The pair-wise overlaps can be used to generate overlap graphs, where each read
and its reverse complement are nodes and edges represent the overlaps between
reads. Edges are weighted by some measure that reflects the strength of an overlap,
such as pair-wise alignments scores combined with the lengths of clone inserts.
3.9.3
Contig Layout Determination.
The problem of creating the fragment layout in contigs, i.e. the problem of determining the relative order and offset of all reads, is based on the reconstruction
model and is NP-complete (Gallant et al., 1980; Gallant, 1983; Kececioglu, 1991).
Hence, all methods rely mostly on heuristics to combine reads. Several sophisticated methods to approximate the optimal solution have been described, for
example, relaxation based on sorting overlaps by decreasing score or spanning
forests (Kececioglu, 1991), methods based on simulated annealing (Burks et al.,
1994), reduction to greedy Eulerian tour (see section 3.8.3), using genetic algorithms (Parsons & E., 1995) or by a series of reductions to an overlap graph that
reduce the number of edges and vertices without changing the space of potential
solutions (Myers, 1995b).
Most of the approximation algorithms perform well on non-repeated data, but have
trouble with the determination of the correct layout in the presence of repeats.
This is true in particular when repeats are highly similar and longer than the
29
average read length. In order to avoid confusing assembly algorithms, several
assembly strategies screen for repeats in the input data set against a database prior
to assembly in order to assemble reads originating from non-repeated regions of
the target sequence into a number of contigs. These contigs may then be ordered
using clone length constraints given by the sequencing of both ends of clone inserts.
This is the approach used in the Celera Assembler (Anson & Myers, 1999; Myers
et al., 2000; Huson et al., 2002). This leaves most of the long repeated regions
unordered. Other ways to detect reads originating from repeated regions of the
target sequence are the computation of a probability of observed coverage, which
is not a precise method, and the detection of conflicting paths in an overlap graph
(Myers, 1995b).
Phrap uses the greedy method, in which the overlaps are first sorted by their
LLR-scores and are joined together into contigs in descending LLR-score order.
Additional information, such as coverage on both strands are used to confirm read
positions. Moreover, the layout may be broken at weak overlaps and the layout recomputed. The sequencing from both ends of an insert yields distance constraints
that can be used both to confirm an assembly, and to aid in the ordering of the
pair-wise overlaps e.g. the TIGR Assembler, CAP3 and ARACHNE, as well as in
the Celera Assembler.
3.9.4
Creation of Multiple Alignments.
In order to get the final result, the consensus sequence, a multiple alignment of
the reads in each contig has to be computed. There are several multiple alignment
methods, that differ mostly by the criteria for selecting pairs of alignments and
thereby the order in which pairs are merged (Gusfield, 1997). However, shotgun
read fragments that are aligned into a contig are assumed to represent the same
region of the target sequence. Therefore the global alignments are most commonly
created by the simple approach of repeatedly aligning the next read with the current alignment, as for example in CAP3. This simple approach results in multiple
alignments that are not locally optimized. In most cases, local deviations in the
multiple alignments will only have a small effect on the quality of the computed
consensus sequence. The multiple alignments can be locally optimized by an algorithm described by Anson and Myers, where each sequence is iteratively aligned
to a consensus structure (Anson & Myers, 1997). After each round of alignments,
the score is computed and compared to the previous score. The iterations are
continued until no lower score can be obtained.
3.9.5
Computation of the Consensus Sequence.
The consensus sequence can be computed using simple majority voting. When
quality values for bases are available, more sophisticated methods may be applied.
The scheme used in CAP3 assumes that the errors are independently distributed
between the strands. For each column of the alignment the quality values for the
base types are partitioned into two groups, one from each strand. The quality
30
values in each group are sorted in decreasing order and the highest quality value
in each group is given the weight of 1.0 and the rest the weight of 0.5. The sum of
quality values is computed. The base with the largest quality value is taken as the
consensus base for the column and the quality of the consensus base is the sum
of quality values for every base minus the sum of the quality values for the other
bases.
Phrap uses an approach where an interval of the best quality read is chosen to
represent the consensus sequence of that interval of the multiple alignment. A
weighted directed graph, where the nodes represent selected read positions and the
bidirectional edges, that represent aligned positions in two overlapping reads with
the weight of zero, is constructed. In addition, the undirectional edges between
positions in the same read with the weight of the sum of the quality values for
the bases between the positions are used. The contig sequence is the path with
the highest weight in the graph. Hence, the method used in Phrap does not yield
a consensus sequence in the proper sense, but is instead a mosaic of the highest
quality regions.
4
4.1
Shotgun Fragment Assembly Programs
Phrap
Several programs are needed to do complete assembly: phred (Ewing et al., 1998)(Ewing & Green, 1998) cross-match, repeat-masker, phrap (http://www.phrap.org) and
consed (Gordon et al., 1998). Phred reads the trace files produced by a sequencing
machine, does base calling and assigns quality values, i.e. error rates for each base.
These quality values reflect the estimated probability that a base is errorneously
called. Cross-match is useful in finding vector sequences in reads.
The assembly is performed by Phrap that uses the quality information provided by
Phred (see section 3.5) to make pair-wise comparisons of reads. This method has
the advantage that it can use almost the whole read length without any significant
trimming, since low quality base calls can be ignored. Phrap makes use of the
information provided by both DNA strands by assuming that the base calls are
independent between the opposite strands. Phrap may confirm a base call by
another call on the opposite strand by summing it with the score of the confirming
call. The same is true for calls performed using different chemistries. Phrap uses
the Phred quality scores to discriminate putative repeats by likelihood ratios. If
mismatches occur along the alignment at positions where base qualities are high,
they are assumed to be due to repeats and reads are not overlapped. If mismatches
occur at low quality positions, they are assumed to be due to base calling errors.
RepeatMasker is a program developed to screen sequences for known repeats to
avoid false overlaps due to repeated regions in the sequenced genomic region.
Detected repeats are masked and not used during the assembly process.
Phrap computes the final sequence or c̈onsensuss̈equence from sections of highest
31
quality regions of the reads in assembled contigs. Consed is used to view the
results of the assembly. Figure 11. shows a screen shot of Consed.
Figure 11: A screenshot of Consed widows: Main, contigview and trace viewer.
4.1.1
The Phrap Method
The hardest step in the assembly process is to determine the correct pattern of
how all the reads align relative to each other (layout). Phrap uses the greedy
algorithm together with log likelihood ratios, LLR scores for overlaps, to determine
the layout of reads. LLR scores are computed using Phred quality values or error
probabilities. The sum of LLR scores are then used to decide if discrepancies in
overlaps are real, or if they are due to base-calling errors.
Candidate overlaps are obtained by dynamic programming and scored initially
without quality values. The scores for these candidate overlaps are re-computed
using error probabilities. The log likelihood ratio H1 /H0 is computed for each base
pair in an overlap and the score of the overlap is the sum of the LLR scores for
each base pair. H1 : the probability of observed data assuming overlap is real. H0 :
the probability observing the data assuming reads are from 95% similar repeats.
The figure 95% is found to work well in practice.
Following assumptions are made:
1. Alignment positions are independent of each other
2. When a base-calling error occurs at a read position, the base calls in the two
reads disagree at the position. (Correction required to avoid this assumption
is very small).
3. Errors in reads are independent of each other.
32
The base pairs in an alignment may match or mismatch. This yields two cases if
possible insertions and deletions are not counted.
Case 1: Bases match.
P rob(agree|H1 (overlap)) = (1 − pi )(1 − pj ).
P rob(agree|H0 (repeat)) = (1 − pi )(1 − pj ) ∗ 0.95
1
LLR = log( 0.95
)
Case 2: Discrepancy between base cells:
P rob(discrep|H1 (overlap)) = pi + pj − pi · pj
P rob(discrep|H0 (repeat)) = 0.05 + 0.95(pi + pj − pi · pj )
LLR ≈ log( pi +pj −pi ·pj 0.05 + 0.95(pi + pj − pi · pj ))
Where pi = probability that the base in sequence i is errorneous and pj = probability that the base in sequence j is errorneous. See figure 12.
Case 1
(i:) A G G G G G G T A C A T C T A G T C A
(j:) T A A T C A G T C A T T G T A
Case 2
Figure 12: Case 1: Bases in the alignment match. Case 2: Bases in the alignment
mismatch.
4.1.2
Greedy Algorithm for Layout Construction
Once all overlaps between reads are computed, the layout, or the order of how the
reads align realitive to each other, can be determined. Phrap maintains the layout
of reads in a graph, where the orientation and offset with respect to the start
of a contig is stored. Vertices in this graph represent reads and edges represent
overlaps. Thus, a contig is the connected component in the graph.
Reads are added to a contig using the greedy algorithm. Initially, each read forms
a separate contig. Each read pair is then processed in decreasing LLR score order.
Each time a read pair is chosen, a potential merge between different contigs is
considered. No negatively scoring read pairs are allowed into a contig and read
pairs must have consistent offset in the candidate contig.
The greedy algorithm together with LLR scores will always result in average correct assemblies. Phrap enchances this result by breaking the layout in positions
33
where reads have positive scores in alternative locations. This time an exhausting
computation is performed, i.e. all possible ways of joining the read fragments is
considered and the highest scoring alternative is chosen.
4.1.3
Construction of Contig Sequences
The final sequence is not strictly a consensus sequence, since the contig sequence is
constructed from the highest quality regions of the reads. The comlumn positions
in a contig are never evaluated. The main steps are as follows:
1. Construct a weighted directed graph:
- Nodes are selected read positions.
- Edges are of two types: 1) Bidirectional: These edges are between
aligned positions in two overlapping reads. The weights of the edges
are zero. 2) Undirectional: These are between positions in the same
read in the 3’ to 5’ direction. Weight of these edges is the sum of Phred
quality measures for bases.
2. Determine the contig sequence from the read segments that lie on the highest
weight path through the graph. The highest weight path can be obtained
using Tarjan’s algorithm with modifications to deal with non-trivial cycles.
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