Scribe: N. Allen

Today's Handouts and Announcements

• No handouts or announcements

Today's Topics

• Fragment Assembly Process
  1. Break DNA molecule into smaller fragments
  2. Sequence the smaller fragments
  3. Assemble smaller fragments to reconstruct original DNA sequence
• Assembly Model: Shortest Common Superstring (SM p. 114)
  • Input: A collection of fragments F.
  • Output: The shortest string S such that for every f in F, S is a superstring of f.
  • This model ignores complemented sequences, sequencing errors, and repetitions.
  • The equivalent decision problem is NP-hard
• Assembly Model: Reconstruction (SM p.116)
  • Input: A collection of fragments F and an error tolerance 0<=e<=1.
  • Output: The shortest string S such that for every f in F, min{d_s(f,S),d_s(f^c,S)} <= e*|f|.
    • d_s is the substring edit distance.
      • d_s(a,b)=min{d(a,s)} where s varies over the substrings of b.
      • d is the classical edit distance.
        • d(a,b) is a measure of the substitutions, insertions, and deletions needed to match a and b.
        • Example: Let x, y, and z be positive real numbers. Then, d(a,b)=x*(# substitutions)+y*(# insertions)+z*(# deletions).
      • Note that d_s is not actually a metric which can cause mathematical and algorithmic problems if not accounted for.
    • f^c is the reverse complement of f. SM uses an overbar to denote this.
  • Handles complemented sequences and sequencing errors but is still an NP-hard decision problem.

• Assembly Model: Multicontig (Class Version) (SM p.117)
  • Input: A collection of fragments F.
  • Output: The layout of F with weakest link of maximal length.
  • A layout is a potential answer that contains every f in F or its complement (exclusively). A layout is thus a multiple alignment of F where every column contains only one kind of base.
    • Example:

      G

      G

      T

      A

      A

      T

      T

      T

      A

      C

      T

      T

      C

      A

      G

      C

      T

      C

      A

      C

      A

      G

      C

  • A link is an overlap of fragments not contained in any other fragment.
  • The weakest link in a layout is the smallest size of any link in it.
  • Another NP-hard decision problem.
• Assembly Model: Multicontig (Book Version) (SM p.117)
  • Input: A collection of fragments F and an integer t.
  • Output: A partition of F into a minimal number of subcollections such that every subcollection has weakest link of length at least t.
• Algorithmic model for overlaps (SM p.119)
  • The overlap multigraph OM(F) of F is a directed, weighted multigraph defined by:
    • The vertex set of OM(F) is F.
    • For every a and b in F, if a != b and there exists some t>=0 such that (ak^t)b=a(k^tb) then there is an edge of weight t from a to b.
  • A simple path in OM(F) corresponds to a superstring of the vertices of OM(F).
  • The length of the superstring will be the sum of the lengths of the fragments minus the sum of the lengths of the edges in the path.
  • Example (continued): TTCAGCTCATTACGGTA
  • Note that the shortest common superstring of the fragments in F will be the Hamiltonian path of maximal weight over OM(F).
• Approximation algorithm for shortest common superstring (SM p.125)
  • Sort the edges of OM(F) by weight.
  • [Iteration] Choose the largest edge satisfying the following properties:
    1. The edge does not form a cycle with previously chosen edges.
    2. The edge does not originate from a vertex that a previously chosen edge originated from.
    3. The edge does not terminate at a vertex that a previously chosen edge terminated at.

Today's Sources

• Setubal and Meidanis: Chapter 4 (pages 114-126).