DNA Fragment Assembly

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1 SIGCSE 009 Algorithms in Bioinformatics Sami Khuri Department of Computer Science San José State University San José, California, USA DNA Fragment Assembly Overlap Graphs Shotgun Sequencing Repeated Regions Sequencing by Hybridization Hamiltonian Cycle Euler Path To Sequence To sequence a DNA molecule is to obtain the string bases that it contains. In large scale DNA sequencing we have to sequence large DNA molecules (thousands of base pairs). Introduction It is impossible to directly sequence contiguous stretches of more than a few hundred bases. On the other hand, we know how to cut random pieces of a long DNA molecule and to produce enough copies of the molecule to sequence. A typical approach to sequence long DNA molecules is to sample and then sequence fragments from them. The problem is that these pieces (fragments) have to be assembled. Steps of Fragment Assembly In large scale DNA sequencing, we are given a collection of many fragments of short DNA sequences. The fragments are approximate substrings of a very long DNA molecule. The consists in reconstructing the original sequence from the fragments. 4.

2 SIGCSE 009 Consensus Sequence Building Importance of Fragment Assembly We need to have reliable, complete genomic sequences of human and other model organisms Base-pair sequence is the most basic piece of DNA information gene structure and function described by sequence Why Sequencing? By comparing genome sequences from carefully chosen organisms, scientists are able to identify specific DNA sequences that have been conserved throughout the evolution of different species, which is a strong indicator that these sequences reflect functionally important regions of the genome. Why Sequencing? (II) To catalog all the genes present in one organism. To compare the gene content of one organism to another organism. To study features other than genes. To study genome evolution. As a foundation for future experimentation. Genome Sequencing Strategies Human Genome Project: map-based strategy individual clones subjected to shotgun sequencing the sequences from the clones (shotgun fragments) then reassembled Celera: whole genome sequence strategy shotgun sequencing On the sequencing of the human genome by Waterston et al., PNAS, vol. 99, 00 4.

3 SIGCSE 009 Hierarchical vs. Whole-Genome Fragments of a DNA Molecule Each fragment corresponds to a substring of one of the strands of the target molecule. We do not know: which strand the fragment belongs to, the position of the fragment relative to the beginning of the strand, if the fragment contains errors. On the sequencing of the human genome by Waterston et al., PNAS, vol. 99, 00 Shotgun Sequencing The Fragment Assembly A large number of fragments are obtained by a sequencing technique: the shotgun method. The reconstruction of the target molecule s sequence is based on fragment overlap. Fragment lengths vary from 00 to 700. Target sequences are between 0,000 and 00,000 base-pairs. The problem consists in obtaining the whole sequence of the target DNA molecule. Since we have a collection of fragments to piece together, this problem is known as the An Example Consider the following four sequences: A C C G T C G T G C T T A C T A C C G T Assume, that it is known that the target sequence is of size 0. The Layout Align in the same column the bases that are equal. Position fragments so that they align well with each other to get a layout. - - A C C G T C G T G C T T A C C T A C C G T - - T T A C C G T G C Consensus of length 9 4.

4 SIGCSE 009 Consensus Sequence The Consensus Sequence or the consensus is obtained by taking a majority vote among all bases in the same column. The answer: TTACCGTGC, has nine bases, close to the approximated value of 0, and contains each fragment as an exact substring. In general, fragments are seldom exact substrings of the consensus. Major Sequencing Centers Joint Genome Institute (USA) Five National Laboratories: Lawrence Berkeley, Lawrence Livermore, Los Alamos, Oak Ridge, Pacific Northwest Stanford Human Genome Center The Institute for Genomic Research (USA) Sanger Institute (UK) J. Craig Venter Institute (USA) Washington University (USA) Integrated Genomics (USA) Genoscope (France) Broad Institute A division of the Whitehead Institute (USA) Chromosome 9 (JGI) It has the highest gene density of all human chromosomes, more than double the genomewide average. It contains,46 protein-coding genes, pseudogenes. Genes that code for such diseases as: insulin-dependent diabetes, myotonic dystrophy, migraines, familial hypercholesterolemia Mimulus guttatus (JGI) Mimulus guttatus: a model organism for studies of evolution and ecology (seep monkey flower) Mimulus species have: a small genome (about 40 Mb), a short generation time (6 - weeks), high fecundity (00 to 000 seeds per pollination), self-compatibility, and ease of greenhouse propagation Complicating Factors DNA sequencing is very challenging since: Real problem instances are very large. Many fragments contain errors: Base call errors Chimeras Vector contamination The orientation of the fragments is frequently unknown; and both strands must be analyzed. There might be a lack of coverage. Repeated Regions Repeats are sequences that appear two or more times in the target molecule. X X Short repeats are repeats covered by one fragment. They do not pose any problem. Long repeats cause most of the problems. 4.4

5 SIGCSE 009 Repeated Regions II Repeats are not necessarily identical, if the similarity is high enough it can be mistaken and considered base call errors. There are two types of repeats: Direct Repeats Inverted Repeats Repeats: An Example Sequence: ATGGCTCATAGGCTCGAG ATGGCTCGAG Repeats: An Example MODELS Sequence: GGCTC TGGCT ATGGC GCTCAT TAGGCT GGCTCG GCTCGA CTCGAG ATGGCTCATAGGCTCGAG GGCTC --GGCTC--- TGGCT -TGGCT---- ATGGC ATGGC----- GCTCAT ---GCTC-AT TAGGCT TAGGCT---- GGCTCG --GGCTCG-- GCTCGA ---GCTCGA- CTCGAG ----CTCGAG GGCTC--- -TGGCT ATGGC GCTCAT TAGGCT GGCTCG GCTCGA CTCGAG ATGGCTCATAGGCTCGAG Models of the fragment assembly problem. Shortest Common Superstring Reconstruction Multicontig None addresses the biological issues completely. Assumption. Fragment Collection is free of contamination and chimeras. Shortest Common Superstring The Shortest Common Superstring (SCS): One of the first attempts to formalize the. Look for the shortest superstring from a collection of given strings. SCS limitations in representing the fragment assembly problem: Does not account for errors. NP hard problem, hence approximation algorithms are used. SCS Problem Definition Input: A collection F of strings Output: A shortest possible string S such that for every f belonging to F, S is a superstring of f. F corresponds to the fragments Each fragment is given by its sequence in the correct orientation S is the sequence of the target DNA molecule. 4.5

6 SIGCSE 009 SCS: An Example Example Let F = {ACT, CTA, AGT} SCS of F, sequence S = ACTAGT S contains all possible fragments in F as substrings. Shortest Common Superstring (SCS) Problem: Given a set of strings, find a shortest string that contains all of them Input: Strings s, s,., s n Output: A string S that contains all strings s, s,., s n as substrings, such that the length of S is minimized. Example: What is the SSP of : {000, 00, 00, 00, 0, 0, 0, }? An Example of SCS SCS: Drawbacks Drawbacks Computational problem specifies S should be a perfect superstring of each fragment. Hence, SCS does not allow for experimental errors in fragments. Orientation must be known, which is seldom the case. Even if the above factors are controlled, it might not be the actual biological solution due to repeated sections in the target DNA sequence. An Introduction to Bioinformatics Algorithms by N. Jones and A. Pevzner FAP Algorithms The Algorithms we consider: Fragments have no errors Fragments are of known orientation Representing Overlays: Common superstring correspond to paths in a graph based on the collection of fragments. Properties of these superstrings are translated to properties of paths It is easier to relate new problems to graphs due to familiarity and knowledge we have about them. Overlap Directed Graphs Given a set F of fragments, we can construct a directed graph as follows: The vertices of F represent the given DNA fragments. If there is an overlap between the suffix of fragment F_ and the prefix of fragment F_, then an edge is drawn from F_ to F_. Each edge is given a weight corresponding to the length of the overlap. 4.6

7 SIGCSE 009 Overlap Graphs Note that the Overlap Graph: Is a multigraph since we can have more than one edge between any vertices in the graph There is an edge between any vertices with weight zero To find the target DNA sequence, we look for a Hamiltonian path: A path that visits each vertex exactly once. We choose the Hamiltonian path with the largest sum of edges. Paths Originating Superstrings Only edges with strictly positive weight are drawn. TACGA Collection F={a,b,c,d} a = TACGA b = ACCC c = d = GACA ACCC GACA Paths Originating Superstrings Collection F={a,b,c,d} a = TACGA b = ACCC c = d = GACA Path P_ = dbc GAGACC Path P_ = abcd TACGACCAGA TACGA ACCC GACA Example: Overlap Multigraph t-overlap: suffix(a,t) = prefix(b,t) Idea: Why don't we search for a path in the overlap multigraph that covers all the vertices? TACGG GGACG GCCC Example: Overlap Multigraph t-overlap: suffix(a,t) = prefix(b,t) Example: Overlap Multigraph t-overlap: suffix(a,t) = prefix(b,t) TACGG TACGG Idea: Why don't we search for a path in the overlap multigraph that covers all the vertices? GGACAG GCCC Idea: Why don't we search for a path in the overlap multigraph that covers all the vertices? GGACAG GCCC 4.7

8 SIGCSE 009 Example: Overlap Multigraph t-overlap: suffix(a,t) = prefix(b,t) Example: Overlap Multigraph t-overlap: suffix(a,t) = prefix(b,t) TACGG TACGG Idea: Why don't we search for a path in the overlap multigraph that covers all the vertices? GGACAG GCCC Idea: Why don't we search for a path in the overlap multigraph that covers all the vertices? GGACAG GCCC Example : Overlap Multigraph F_ = F_ = F_ = F_4 = F_5 = Reconstruct the target DNA sequence from the given fragments 4.8

9 SIGCSE ACCGCATGACCACTA Shortest Superstrings As Paths A collection F is said to be substring-free if there are no two distinct strings a and b in F such that a is a substring of b. Let F be a substring-free collection. Then for every common superstring S of F there is a Hamiltonian path P in OM(F) such that S(P) is a subsequence of S. Let F be a substring-free collection. If S is a shortest common superstring of F, there is a Hamiltonian path P such that S=S(P) The Overlap Graph Looking for shortest common superstrings is the same as looking for Hamiltonian paths of maximum weight in a directed multigraph. Goal: maximize the weight. Simplify the multigraph, consider only the heaviest edge between every pair of nodes, discarding other parallel edges. Call the new graph the overlap graph of F denoted by OG(F). 4.9

10 SIGCSE 009 The Greedy Algorithm Edges are processed in non increasing order by weight. Continuously add the heaviest available edge as long as it does not upset the construction of the Hamiltonian path given the previously chosen edges. The procedure ends when there are exactly n- edges, or when the accepted edges induce a connected subgraph. Example: Greedy Algorithm Fails F={ATGC, GCC, TGCAT} ATGC GCC TGCAT Order the edges by weight (ATGC, TGCAT) = (ATGC, GCC) = (TGCAT, ATGC) = The greedy algorithm will choose first (ATGC, TGCAT) = and then is forced to select an edge with weight 0 to complete the path: (ATGC, TGCAT) (TGCAT,GCC) Instead the solution should be (TGCAT, ATGC) = (ATGC, GCC) = Sequencing by Hybridization Sequencing by Hybridization Universal DNA Array detects all the k-mers in given DNA sample (red dots) Genome Sequence Assembly by Mihai Pop, TIGR Spectrum ( T, l ): The set of all possible (n l + ) l-mers in a string T of length n The order of individual elements in Spectrum ( T, l ) does not matter Example: T = ATGCGTGGCA Spectrum (T, ) = {ATG, TGC, GCG, CGT, GTG, TGG, GGC, GCA} The SBH Problem SBH: An Example (I) Goal: Reconstruct a string T from its l-mer composition Input: A set S, representing all l-mers from an (unknown) string T Output: String T such that Spectrum(T,l) = S S = {ACG,CGC,GCA,CAT,ATC} hybridization A C G G C A C A T A T C Spectrum for k= DNA Sample A A T C Adapted from Shuai Cheng Li: CS48/68 4.0

11 SIGCSE 009 SBH: An Example (II) S = {ACG,CGC,GCA,CAT,ATC} DNA Sample hybridization A A T C Two Samples, One Spectrum Two samples may result in the same spectrum More information is needed to construct a unique sequence A C G G C A C A T A T C Spectrum for k= A C G G C A C A T A T C A A T C T is such that Spectrum (T, ) = {ACG,CGC,GCA,CAT,ATC} In other words, Spectrum(T,) = S Adapted from Shuai Cheng Li: CS48/68 T A C C T C C A A C C C C T C T C T C C C C G G C C C C A A C C T C C A A C C C C G G C C C C T C T C T C C C C A Adapted from Shuai Cheng Li: CS48/68 Two Samples, One Spectrum (II) Two samples may result in the same spectrum More information is needed to construct a unique sequence A C C T C C A A C C C C T C T C T C C C C G G C C C C A A C C T C C A A C C C C G G C C C C T C T C T C C C C A Adapted from Shuai Cheng Li: CS48/68 SBH and Eulerian Path Given a spectrum S, draw a directed graph where: Each vertex represents a (k-)-prefix or (k-)-suffix of k-mers in S Each edge is a k-mer from S connecting a vertex representing a (k-)-prefix and a (k-)-suffix. Find a Eulerian path of G, and reconstruct the sequence from the path Example: Spectrum= {ACG, ATC, CAT, CGC, GCA} Edges: ACG, ATC, CAT, CGC and GCA Vertices: AC, CG, AT, TC, CA, and GC. Adapted from Shuai Cheng Li: CS48/68 Eulerian Path: An Example Example: Spectrum= {ACG, ATC, CAT, CGC, GCA} Draw the vertices: AC, AT, CA, CG, GC, TC (alphabetical order) SBH and Eulerian Path (II) Example: Spectrum= {ACG, ATC, CAT, CGC, GCA} Draw the vertices: AC, AT, CA, CG, GC, TC (alphabetical order) Draw edge from vertex AC to vertex CG edge ACG ACG AC AT CA CG GC TC AC AT CA CG GC TC Adapted from Shuai Cheng Li: CS48/68 4.

12 SIGCSE 009 SBH and Eulerian Path (III) Example: Spectrum= {ACG, ATC, CAT, CGC, GCA} Draw the vertices: AC, AT, CA, CG, GC, TC (alphabetical order) Draw edge from vertex AC to vertex CG edge ACG Draw edge from vertex AT to vertex TC edge ATC ACG SBH and Eulerian Path (IV) Spectrum= {ACG, ATC, CAT, CGC, GCA} Draw the vertices: AC, AT, CA, CG, GC, TC (alphabetical order) Draw edge from vertex AC to vertex CG edge ACG Draw edge from vertex AT to vertex TC edge ATC Draw edge from vertex CA to vertex AT edge CAT Draw edge from vertex CG to vertex GC edge CGC Draw edge from vertex GC to vertex CA edge GCA AC AT CA CG GC TC AC AT CA CG GC TC ATC SBH and Eulerian Path (V) An Eulerian Path is a path which visits each edge of the graph once Eulerian path: ACCG GC CA AT TC Sequence: ACGCATC Multiple paths are possible Uniqueness Spectrum={ATG, TGC, GCG, CGT, GTG, TGG, GGC, GCA } GT CG GT CG AT TG GC CA AT TG GC CA GG GG AC AT CA CG GC TC ATGCGTGGCA ATGGCGTGCA Adapted from Shuai Cheng Li: CS48/68 Challenges of SBH The solution may not be unique For example: Obtain an Eulerian cycle instead of a path multiple solutions The input data, the Spectrum S, may contain errors For example: false positives, false negatives, uncertain frequency of k-mers Multiple parallel edges ambiguous solutions Some Solutions Several solutions were proposed to solve the problems Positional Eulerian Path (PEP) by Hannnenhalli et al. 996 Positional Sequencing by Hybridization (PSBH) add extra information to probes Interactive Protocols by Skiena et al. 995 Gapped probes by Preparata et al. 000 and Frieze et al. 999 Analog-Spectrum by Preparata 004 Note that we consider the simple case were the spectrum yields an Euler path. 4.