CSE 746

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• CSE 746 Introduction to Bioinformatics
• Research Project
• Two methods of DNA Sequencing Comparing and
  Intertwining Suffix Trees and De Bruijn Graphs
  for Sequence Assembly
• Dicle Öztürk
• 540110004

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Suffix trees - Definition

• Definition (Gusfield)

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Suffix trees uses and complexity

• Useful in string search, text processing, tasks
• More like bridge between exact and inexact
  pattern matching problems
• Storing suffix trees requires more space than
  storing the string itself
• It was Ukkonen who was the first to provide a
  linear time online construction of suffix trees
• Can be used for the sorting stage of BWT

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Suffix trees Naïve algorithm

• Assuming a bounded alphabet, this algorithm runs
  in O(m2) time.
• --
• N1 root (initially a leaf)
• Ni assumption
• Ni1 inductive string constructed using Ni

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Suffix Trees Naïve algorithm

• find the longest path from root whose label
  matches a prefix of Si1...m
• (Matching path is unique because no two edges out
  of a node can have labels that begin with the
  same char)
• if no further match is possible
• if in the middle of an edge (u,v)
• split the edge into two
• insert a new node w just after the last
  char on the edge that matched a char in
  Si1...m (before the char mismatched)
• label the edges (u,w) and (w,v)
  accordingly
• endif
• create a new edge (w,i1), thus creating a
  new leaf (i1)
• label (w,i1) with the unmatched part of the
  suffix Si1...m
• endif

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Suffix trees Ukkonens algorithm

• Ukkonen moves Mccreights work further,
  decreasing space complexity to linear and giving
  comprehendible definitions.
• The tree is constructed online and in a
  left-to-right fashion, as opposed to Weiners
  method.

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Suffix trees Ukkonens algorithm

• In Ukkonens algorithm, substrings are kept by
  their indices.
• The trick is that the last index of suffixes are
  not defined, which are represented by leaves.
• If w is a substring of the string s, w(i,j) is
  actually w si...sj.
• Thus, the suffix tree for s will have at most s
  leaves, guaranteeing linear complexity in space.

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Suffix trees - Applications

• In the notes of (Lewis, usask.ca), some general
  applications of suffix trees in computational
  biology are mentioned,
• Genome alignment
• Signature selection
• Finding a short sequence that is specific to
  individual genes
• Searches for non-repeating segments
• Finding an representing all tandem repeats

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Suffix trees - Applications

• (Riedl, 1994) gives a more detailed list of
  applications,
• Suffix trees are useful on search, single
  sequence analysis and multiple sequence analysis
• With the method they use, which is called Gestalt
  tree matching, homology-search applications are
  believed to outperform fastp and fasta

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Suffix trees - Applications

• Detection and occurrences of any number of short
  subsequences can be useful in enzyme cut-site
  determination
• Generalised suffix tree of a set of sequences
  allows all of the sequences to be analysed
  simultaneously
• Detection of common subsequences within a set of
  sequences can be applied to contig reassembly
  (Riedl, 1994)

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Suffix trees - Applications

• Finding the best match between the suffix of one
  read and the prefix of another can also be a
  fruitful task
• Suffix-prefix overlaps can help for finding the
  shortest common superstrings of reads, especially
  in genome assembly
• Suffix trees can be used to remove redundancies
  in string containment problems

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De Bruijn Graphs

• There exist strings which are called De Bruijn
  strings, which might have given some inspiration
  to the development of De Bruijn graphs and vice
  versa.
• A De Bruijn string of order k is a non-empty
  string x which is defined over an alphabet A (
  x?A) such that if each string on A of length k
  occurs once and only once in x. Like x11001
  where A 0,1.

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De Bruijn Graphs

• De Bruijn graphs models those kinds of strings
  where the nodes hold the substrings of length k-1
  and edges have one character (leftover of
  k-length substring). If the two nodes are
  connected by an edge, the one being the source
  follows the other (it is a directed graph).
• Building De Bruijn graphs is not a piece of cake
  but it has many applications in genome assembly

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De Bruijn Graphs

• De Bruijn graphs are useful in
• Handling sequence variants like duplications,
  inversions and transpositions
• Combining sequences if different length
• Effective data compression even when the data has
  many redundant parts
• Detecting and analysing structural variants from
  unassembled data

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De Bruijn Graphs and Affix Trees

• Conceptually, the De Bruijn graph of a sequence
  can be considered as a simplification of that
  sequence's affix tree.
• Each non-empty substring of a given sequence is
  mapped onto a separate node
• Each node is connected by an edge to its longest
  prefix and by a suffix link to its longest suffix
• Nodes corresponding to sequences of length 1 are
  directly connected to the root node,
  corresponding to t.

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De Bruijn Graphs and Affix Trees

• Root represents empty string. The first children
  are the sequences of length 1.
• The analogy is built upon the atomic tree
  representation of (Giegerich, 1997) and the idea
  is mostly from (Maaß, 2003).
• It has been pointed out in (Zerbino, 2009) that
  traversing the De Bruijn graoh is equivalent to
  traversing the affix tree across its breadth

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De Bruijn Graphs and Affix Trees

• Furthermore it says,
• If we rank the nodes by distance from the root,
  the k-mer nodes of the De Bruijn graph correspond
  to the nodes of rank k in the affix tree
• It is easy to demonstrate that two k-mers are
  connected in the De Bruijn graph iff the
  corresponding nodes in the affix tree are
  connected by a path composed of an edge and a
  suffix link, going through a node of rank k1

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De Bruijn Graphs and Affix Trees

• (Giegreich, 1997) gives the definitions of
  suffix and prefix trees together with their
  special relationship.
• Active suffixes and prefixes for the string t
• The active suffix of t ? its longest nested
  suffix denoted as a(t)
• The active prefix of t ? its longest nested
  prefix denoted as a-1(t)
• Then, a(t-1) (a-1(t))-1

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De Bruijn Graphs and Affix Trees

• The tree is atomic of each of its edges is marked
  by a single char
• So every node is explicit
• The tree is actually a trie

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De Bruijn Graphs and Affix Trees

• (Maaß, 2003) furthers this analogy-based idea of
  affix trees and gives some more insight into the
  issue
• It says,
• A suffix link is an auxiliary edge from node n to
  node m where m is the node such that path(m) is
  the longest proper prefix of path(n) represented
  by a node in the tree.
• Suffix links are used to move from one node to
  another so that the represented string is
  shortened at the front.
• It mentions also that in essence, it was actually
  (Blummer, 1998) who observed the dual structure
  of suffix trees.

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Redundancies in the sequences

• And finally, we can say that the redundancy in
  the suffix tree of some string should be as less
  as possible so that an efficient build-up of De
  Bruijn graph out that tree can be obtained.
• To reduce redundancy, some compression methods
  can be applied but no loss should take place and
  reversal should be possible. The algorithm of
  Lempel and Ziv is advised to be an efficient tool
  for this task, running in O(n) time with suffix
  trees.

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References

• 1 Algorithms on Strings, Trees, and
  Sequences Computer Science and Computational
  Biology, Dan Gusfield, Cambridge University
  Press, Jan 15, 1997.
• 2 Ukkonen E., On-line Construction of
  Suffix-Trees, Algorithmica vol 14(3), 1995.
• 3 - Algorithms on Strings, Maxime Crochemore
  and Christophe Hancart, Cambridge University
  Press, June 2007.
• 4 Genome assembly and comparison using de
  Bruijn graphs, D.R. Zerbino, PhD Thesis, European
  Bioinformatics Institute, Darwin College,
  September, 2009.
• 5 Giegerich R., and Kurtz S., From Ukkonen to
  McCreight and Weiner A unifying view of
  linear-time suf?x tree construction, Algorithmica
  19331353, 1997
• 6 Maaß, M. G., Linear bidirectional on-line
  construction of af?x trees, Algorithmica vol.
  37(1), 2003.
• 7 Bieganski, P., Riedl, J., Cartis, J.V.,
  Retzel, E.F., Generalized Suffix Trees for
  Biological Sequence Data Implementations and
  Applications, HICSS (5), 1994.