Assembling%20Algorithms%20and%20Techniques

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Assembling Algorithms and Techniques

• Upmanyu Misra
• Computational Issues in Molecular Biology
• CSE 397-497

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Motivation

• Direct sequencing of full stretches of DNA base
  pairs is not possible (as of now)
• Hence, we sample DNA stretches by breaking them
  in fragments
• Sequencing of fragments is done by previously
  discussed algorithms
• Fragments are to assembled back to make sense
  of the full DNA stretches

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Approach

• Primitive (Root) Techniques
• Models used
• Algorithms Greedy/Multicontig
• Heuristics
• Greedy Path Merge Heuristics
• Basics/Examples/Definitions wherever required

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Refreshing Basics

• Consider 4 strings (fragments) to be assembled
• CTAGG
• AGGTA
• GAACCT
• TAAC
• We know that final sequence size should be around
  10 base pairs

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How Assembling works

• _ _ C T A G G _ _
• _ _ _ _ A G G T A
• G A C T A _ _ _ _
• _ _ _ T A G G_ _

G A C T A G G T A
CONSENSUS
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Trivial, eh?

• Consensus sequence easily achieved
• Very close to our goal of finding a 10 base
  pair chain
• A combination was found such that each column was
  coherent
• The consensus contained each fragment as an exact
  substring

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Size does matter

• Errors creep in when we consider the real life
  scenario
• Size of sequence increases
• Size of fragments increases
• Noise
• Loss of data
• Larger consensus goals

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Pandoras Box

• Errors might be of the following type (and many
  more)
• Base Call Insertion Deletion - Substitution
• Chimeras
• Unknown Orientations
• Repeated Regions
• Lack of coverage

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Primitive Solutions

• Assembling for Shotgun fragments
• Directed Sequencing or Walking
• Dual End Sequencing
• Non-Shotgun (based on experimental data)
• Sequencing by Hybridization
• Good news Provide headway into the assembly
  problem
• Bad news None of them, by itself, is good
  enough, not even close

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Usable Models
Real World Problems Contamination Chimras Experiment Errors Orientation Errors Repeated Sections Lack of Coverage
Models Contamination Chimras Experiment Errors Orientation Errors Repeated Sections Lack of Coverage
Shortest Common Superstring
Reconstr -uction
Multicontig
NP HARD
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Algorithms

• Again, these algorithms are of little value by
  themselves. Good supporting heuristics help to
  make them usable
• Two different approaches
• GREEDY
• MULTICONTIG (acyclic overlap graphs)

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Getting Started

• Linking overlaps to graphsWhy?
• Superstrings can be matched on to Graphs
• We (the Computer Science community!) come across
  Graphs frequently
• Comfortable (logically, psychologically,
  cognitively)

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Definitions

• def. Overlap Multigraph OM(F) of a collection F
  is the directed, weighted. The set V of nodes is
  the set F. A directed edge from a ? F to a
  different fragment b ? F with weights t 0
  exists if a suffix of a with t characters is a
  prefix of b.

suffix(a,t) prefix (b,t)
k is the Killer Agent such that kATCG TCG, k
-1
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Hence...
TACGA
Consider F is a,b,c,d a TACGA b ACCC c
CTAAAG d GACA
t 2
GACA
1
a
b
2
1
1
c
d
1
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Properties of the Graph

• Node must not have a path to itself
• We keep the zero weight edges too
• (hence n(n-1) edges)
• We denote path P1 dbc such that
• G A C A _ _ _ _ _ _ _
• _ _ _ A C C C _ _ _ _
• _ _ _ _ _ _ C T A A G

a TACGA b ACCC c CTAAAG d GACA
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Properties of the Graph

• If A is the set of fragments involved in P, we
  will have exactly edges in P
• Common superstring will be called S(P)
• Relationship between total length of A, the
  paths weight and the superstring length is,
• A w(P) S(P)

A 1
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Picture is worth .
G A C A _ _ _ _ _ _ _ _ _
_ _ A C C C _ _ _ _ _ _ _ _ _ _ _
C T A A A G
2 w(P)

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14 A
S(P) G A C A C C C T A A A G
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Mathematical Representation

• OR, we may say,
• If

computing length
A - w(P)
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Hamiltonian Path

• Each path we traverse gives a common superstring
  of the fragments involved
• If there is a path that passes through each
  vertex, we have a common superstring containing
  all fragments of F
• Hence S(P) F - w(P)
• We have been talking about
• Observation Since F is constant, minimizing
  S(P) is equivalent to maximizing w(P)

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Hamiltonian Graph

http//mathworld.wolfram.com/HamiltonianGraph.html
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Backtracking

• Every path corresponds to a superstring
• Is the converse true?
• Consider shortest superstrings
• YES, they will always correspond to a path

NO
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Important results

• def. A collection of fragments is
  substring-free if there are no two distinct
  strings in the collection such that any one of
  them is a substring for the other
• For example,
• F GAA, CTGA, AGTCACGCAA is SFree
• but,
• F ACTGAA, TGCA, CTGA, ACGA is not

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Important results

• THEOREM 1 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
• Example F tag, cat, gac
• S(P) tagacat / tagcatgac / catagac /
• S any superstring (with/without faulty
  characters)

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Important results
TAG
GAC
1
1
1
1
0
0
CAT
S(P) tagacat / tagcatgac / catagac /
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Important results

• def. F dominates another collection G if every
  elements of G is a substring of some element of
  F.
• For eg.,
• F ATGA, CATAGAA, GTACTAA
• G TAG, CAT, TACT

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Important results

• def. F and G are equivalent if F dominates G
  and G dominates F
• F TGA, CTTGAA, ACTTGAA
• G TTGA, ACTTGAA

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Important results

• LEMMA Two equivalent substring-free collections
  are identical
• F TGA, CTTGAA, ACTTGAA
• G TTGA, ACTTGAA
• F ACTTGAA
• G ACTTGAA

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Important results

• THEOREM 2 F is a collection of strings. There is
  a unique substring-free collection G equivalent
  to F
• This result implies that if we are looking for
  common superstrings, then we just have to look
  for substring-free collections, since every
  collection will have one equivalent to it.
• Hence, just removing all strings from F that are
  substrings of other elements in F solves our
  purpose

Hamiltonian Path SCS SCS
Substring free
Hamiltonian Path
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GREEDY ALGORITHM

• Looking for shortest common superstrings is same
  as looking for Hamiltonian paths of maximum
  weights in a directed multigraph
• Since only heaviest edges are required, we may
  prune the weaker ones, saving time and space
• This is a greedy attempt

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GREEDY ALGORITHM

• Lets look at an example
• Considering previous graph

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Hence...
TACGA
Consider F is a,b,c,d a TACGA b ACCC c
CTAAAG d GACA
t 2
GACA
1
a
b
2
1
TACGA
1
GACA
c
d
ACCC
1
CTAAAG
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IMPLEMENTATION

• Graphs are easier to understand, but not
  necessary
• We can implement the greedy algorithm by
  recursive implementation of following procedure,
  until only one fragment remains.

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IMPLEMENTATION

• Take pair (f, g) of fragments with largest
  overlap, say T.
• Remove both fragments from F and add fkTg to F
• Assumption, F is substring-free

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Greed seldom pays
Consider F is a,b,c a ATGC b TGCAT c GCC
2
3
a
b
2
0
c
Greedy Algorithm follows the path a ? b ? c
giving a total weight of 3 including a zero
weight path
The Best path would have been b ? a ? c giving a
weight of 4
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HEURISTICS

• Motivation
• - Algorithms like Greedy do not
  guarantee optimal solutions
• - Solving Shortest Common Superstrings
  through Hamiltonian paths is NP-Complete
• - Closeness of problem to Multiple alignment
  problem

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HEURISTICS

• Following heuristics mostly aim towards solving
  two major problems
• Fragments can participate with either direct or
  the reverse-complemented sequence
• Fragments themselves are usually much shorter
  than the alignment

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HEURISTICS

• calls for discrimination between treatment of
  different types of gaps (internal, external), to
  bind the fragment characters
• urges us to consider other criteria, besides the
  score, to assess the quality of an alignment

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Accessing Criteria

• SCORING
• Measure of participation of each aligned column
  in a multiple alignment
• Entropy may be used for that

• Entropy measures the uniformity of alignment

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Accessing Criteria

• COVERAGE
• A fragment covers a column i if it participates
  in this column either with a character or with an
  internal space
• Minimum, maximum and mean coverage can be
  calculated for a layout
• Lesser the coverage, weaker the connection and
  more the independence
• Coverage bolsters/weakens the notion of consensus
  sequence

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Accessing Criteria

• Linkage
• Measure of the way individual fragments are
  linked to each other in the layout
• Fragments should have overlapping ends to show
  some linkage

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Summing up

• For practical implementation we may divide the
  process in three phases
• Finding overlaps
• Building a layout
• Computing the consensus

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Greedy Path Merge Algorithm
Clone-by-Clone Approach (by Human Genome project)
Whole Genome Shotgun (Celera Genomics)
Preliminary Data
Heuristic Greedy Path Merge
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Greedy Path Merge Algorithm

• HGPs clone-by-clone approach
• Constructs a tiling of the genome by overlapping
  pieces ( 150k bp)
• Concentrates on determining the sequence of each
  such piece
• Pieces are called BAC clones or simply BACs
  (Bacterial Artificial Chromosomes)
• BAC is randomly broken into many smaller
  fragments that are cloned and sequenced
• The fragments are assembled resulting in Contigs

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Greedy Path Merge Algorithm

• WGS strategy
• Whole Genome is randomly broken into smaller
  pieces that are sequenced
• Due to size of Genome a pure overlap-based
  approach is not feasible
• Additionally, Celera uses mate-pairs. These are
  fragments with a known relative distance and a
  standard deviation (sort of experimental data
  from sampling and sequencing large chunks of DNA

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Greedy Path Merge Algorithm

• def. Bactigs are pieces of DNA sequences that
  are obtained from a common source region of
  150k bp contiguous DNA of human genome obtained
  by shotgun sequencing and assembly
• BACs start as phase-0 BAC, which means several
  bactids of small size. They evolve upto phase-3
  BAC, which is one bactig fully representing its
  source sequence

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Greedy Path Merge Algorithm

• This approach takes up the BACs and Celeras
  fragments and primarily aims at increasing the
  level of assembly of BACs using the information
  given by fragments and mate links
• Hence evolving phase-1/2 BACs to phase-3 BACs

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Greedy Path Merge Algorithm

• INITIAL BACTIG GRAPH
• BAC BB1, B2, .., Bn. Fragment f hits (or is
  embedded in) a bactig Bi if it (or its reverse
  complement) aligns with the bactig with a high
  density
• Bactig graph is a weighted, undirected
  multi-graph without self-loops

Mate edge
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Greedy Path Merge Algorithm

• Functions performed on Initial Bactig Graph
• Edge Bundling For more than one mate pairs
• Transitive Reduction Transitively resucing long
  mate edges
• Hence Final Bactig Graph is the initial one
  with Edge Bundling and Transitive Reduction
  performed on it

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Greedy Path Merge Algorithm

• BACTIG ORDERING PROBLEM
• Given a Bactig graph G. To find an ordering of G
  that maximizes the weights of valid (happy) mate
  edges
• The problem is NP complete
• From here on we proceed almost exactly as in
  normal Greedy Algorithm

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Greedy Path Merge Algorithm

• Throughout the implementation it is maintained
  that every node is adjacent to at most two
  selected edges. These edges form the selected
  path.
• The ordering of Bactigs induced by such a path is
  called Scaffolding of Bactigs
• The deviation from the actual algorithm is that
  it can introduce new edges to the bactig graph,
  which are called inferred edges

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Greedy Path Merge Algorithm

• Given a bactig graph G. The output of the
  algorithm is a node-disjoint covering of G by
  selected paths, each one defining an ordering of
  the bactigs whose edges it covers
• The algorithm runs in O(mnm2) time, where m is
  he number of mate pair edges and n is the number
  of bactig edges

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