Fragment Assembly

1
Fragment Assembly

2
Introduction

• Fragments are typically of 200-700 bp long
• Target string is about 30k 100k bp long
• Problem given a set of fragments reconstruct the
  target

3
Introduction

• Multiple-alignment of the fragments ignoring
  spaces at the end
• The alignment is called layout
• The output is called the consensus sequence
• An optimization problem

4
Complications

• Base-call errors
• Substitution errors p 107
• Insertion errors (possibly from the host
  sequence) p 108, fig 4.3
• Deletion error fig 4.4
• Majority voting solves them (or some form of
  optimization)

5
Complications

• Chimeras
• To non-contiguous fragments get joined as a
  single fragment p 109, fig 4.5
• Needs to be weeded out as a preprocessing step
• Similar to chimeras, contaminant fragments
  (possibly from host) needs to be filtered out as
  well

6
Complications

• Unknown orientation
• Fragments may come from either strand
• Even from the opposite strand, its
  reverse-complement must be in the target string
• Consequence try both forward and rev-complement
  of each fragment (2n trial in worst, for n
  fragments)
• p 109, fig 4.6

7
Complications

• Repeats
• Regions (super-string of some fragments) may
  repeat in a target
• Consequent problem where do the fragments really
  come from, on approximate alignment? p 110, fig
  4.7
• Problem 2 where should the inter-repeat
  fragments go? p111, fig 4.8, fig 4.9
• Inverted repeats repeat of the reverse
  complement fig 4.10

8
Complications

• Insufficient coverage
• Chance of coverage increases with redundancy (a
  heuristic cover 8 times the target length)
• Chance of covering a gap reduces when it remains
  uncovered even after multiple fragments are
  aligned) random sampling is not good solution
  here

9
Complications

• Insufficient coverage
• What you get with insufficient coverage is
  multiple contigs, not one contig
• t-contig is where we expect t-long overlap
  between pairs of fragments
• Expected number of contigs p 112, formula 4.1
• Lower t means lesser number of contigs (more
  aligned segments), but weaker consensus

10
Reconstruction

• Shortest common superstrings are not the best
  solution
• Fig 4.12 vs Fig 4.13 (p115/116)

11
Reconstruction

• Superstring to be reconstructed out of fragments
• An alignment problem with no end penalty
• d_s is edit distance score without end-penalty
  minimized over edit distances d
• Fig 4.14 (p117) for best aligned
  subsequence-matching
• Note, char matched is charged 0, mismatch 1, gap
  2, in distance rather than similarity
• We will use d for d_s

12
Reconstruction

• f is approximate substring of S at error level e,
  then the score is
• d(f, S) lt ef,
• e1 means no error allowed
• elt1 allows insert/delete/substitution errors
• f and f- both should be matched

13
Reconstruction Problem

• Input Set F of substrings, error level e
• Output Shortest possible string S s.t. for all f
  
• Min(d(f, S), d(f-, S)) lt ef

14
Reconstruction Multicontig

• How much overlap do we require between strings?
• Ideally, each column in the layout L should have
  same character, for all columns 1 through L
• Fig 4.4 (p 118) t-contig for t3, 2, 1
• Balance between t and number of t-contigs

15
Reconstruction Multicontig

• S is e-consensus sequence (multicontig) for
  0ltelt1 edit distance d(f, S) lt ef
• Multicontig problem
• Input set F, integer tgt0, 0ltelt1
• Output Minimum partition over F, each partition
  Ci is a t-contig with e-consensus

16
Reconstruction Overlap Multi-graph

• Nodes are the fragments
• Directed arcs label length t of overlap between
  nodes t-suffix t-prefix
• Arcs between all pairs of nodes, but no self-loop
• Fig 4.15 (p 121) example
• Length of a created superstringtotal wt along
  the path(or overlaps) total length of all
  fragments involved
• Max weight Hamiltonian path is what we are
  looking for in this graph ? max overlapped
  superstring

17
Reconstruction

• Substrings of fragments within the set of
  fragments are noise remove them
• Draw OMG of the substring free set of fragments
• Shortest common superstring always correspond to
  a Hamiltonian path in this graph

18
Reconstruction OMG

• Thm 4.1 (p 123) F substring free, for every
  common superstring S, there is a Ham. Path P,
  s.t., S(P) is in S
• Substrings are strictly ordered over S order of
  left pts order of rt points (otherwise
  substring exists)
• Path follows the same order of fragments (as in
  S) in OMG
• S may contain extra garbage materials, so, S(P)
  is within S

19
Reconstruction OMG

• If S is shortest common superstring, then S must
  be within S(P), or SS(P)
• In other words, a Ham. Path in OMG for
  substring-free collection F is a shortest common
  superstring of the Fragment set F

20
Reconstruction OMG

• Think of an algorithm for weeding out substrings
  from F
• Also, weed out multi-edges by keeping the largest
  wt edge between any pair of nodes
• If the wt on an edge is below a threshold t, then
  the wt should be treated as 0

21
Reconstruction OMG

• Greedy Algorithm to draw Ham. Path (p 125)
• Collects edges largest to smallest,
• (1) preventing cycle (union-find),
• (2) indegree of each node should be lt1 (first
  node has 0)
• (3) outdegree of each node should be lt1 (last
  node has 0)
• Does not return Ham. Path. Can you modify to
  return Ham. Path?
• Alg is NOT optimal, example (p 126) returns 3,
  optimal wt is 4

22
Reconstruction OMG

• Subintervals if a fragment can be embedded
  within another one in the set
• Subinterval-free and repeat-free graphs connected
  at level t has a Ham. Path that generates the
  target string

23
Reconstruction OMG

• If a repeat exists in the original string, then
  the graph will have a cycle
• False positive substrings from two different
  portions has t-overlap
• If a cycle exist in the graph, then there must be
  a false positive (Thm 4.4, p129) proof by
  contradiction, otherwise the subinterval-free
  fragments can be totally ordered

24
Reconstruction OMG

• If there is no repeats in a subinterval-free
  graph, then there exist a unique Ham. Path
• If there exist a cycle it may not come from a
  repeat

25
Reconstruction OMG

• Example 4.6 (p 130) greedy alg finds wrong
  string, but the Ham. Path finds the correct one
• Greedy does not care about linkage (optimizes on
  total overlap finds shortest common
  superstring)
• Ham path chooses any t-overlap connections
  cares for linkage only

26
Parameters in aligning for fragment assembly

• Score on a column traditionally 0,-1,-2 in
  sum-of-pairs
• Entropy
• Sumover alphabets and space c pc log pc,
  where pc is probability of c
• All same character, pc 1, entropy0
• For a, t, c, g, -, all different, pc 1/5,
  entropylog 5entropy measures uniformity alone, a
  better metric

27
Parameters in aligning for fragment assembly

• Coverage How many each column is covered by
  how many fragments? (Average, min, max)
• This is different from the concept of t-overlap
• If a column (of the target) is covered by 0, then
  the layout is disconnected
• Counteracts with the requirement of
  subinterval-free collection if we expect
  coveragegt1 for all columns

28
Parameters in aligning for fragment assembly

• Coverage is not enough, we need good linkage,
  Example p 133
• Ham. Path algorithm is doing that

29
Steps in assembly

• Step 1 Overlap finding
• Approximate delete, insert, replace allowed
• by semi-global DP algorithm
• with appropriate end-gap penalty,
• pairwise between each fragment and its
  reverse-complement

30
Steps in assembly

• Step 2 Construct over (F union F-bar) for the
  fragment set F
• (-- after eliminating substrings?)
• Construct Hamiltonian path in this graph
• Cycles and unbalanced coverage may mean repeats

31
Steps in assembly

• Step 3 fine tuning the multiple alignment to get
  a consensus target
• Manual or algorithmic
• Examples in p 137-138