1. Fragment Assembly
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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
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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
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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)
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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
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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 (2^n trial in worst, for n fragments)
[p 109, fig 4.6]
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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]
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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
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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
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10. Reconstruction Shortest common superstrings are not the best solution
Fig 4.12 vs Fig 4.13 (p115/116)
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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
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12. Reconstruction
f is approximate substring of S at error level e, then the score is
d(f, S) =< e|f|,
e=1 means no error allowed
e<1 allows insert/delete/substitution errors
f and f- both should be matched
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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)) =< e|f|
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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 t=3, 2, 1
Balance between t and number of t-contigs
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15. Reconstruction: Multicontig
S is e-consensus sequence (multicontig) for 0=<e=<1: edit distance d(f, S) =< e|f|
Multicontig problem:
Input: set F, integer t>=0, 0=<e=<1
Output: Minimum partition over F, each partition Ci is a t-contig with e-consensus
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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 superstring=total 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
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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
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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
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19. Reconstruction: OMG
If S is shortest common superstring, then S must be within S(P), or S=S(P)
In other words, a Ham. Path in OMG for substring-free collection F’ is a shortest common superstring of the Fragment set F
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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
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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 =<1 (first node has 0)
(3) outdegree of each node should be =<1 (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
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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
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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
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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
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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
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26. Parameters in aligning for fragment assembly
Score on a column: traditionally {0,-1,-2} in sum-of-pairs
Entropy:
Sum[over alphabets and space c] –pc log pc, where pc is probability of c
All same character, pc = 1, entropy=0
For {a, t, c, g, -}, all different, pc = 1/5, entropy=log 5entropy measures uniformity alone, a better metric
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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 coverage>1 for all columns
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28. Parameters in aligning for fragment assembly
Coverage is not enough, we need good linkage, Example: p 133
Ham. Path algorithm is doing that
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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
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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
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31. Steps in assembly :
Step 3: fine tuning the multiple alignment to get a consensus target
Manual or algorithmic
Examples in p
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