Combinatorial Optimization Problems in Computational Biology

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Combinatorial Optimization Problems in
Computational Biology

• Ion Mandoiu
• CSE Department

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What Is Computational Biology?

• G. Lancia Study of mathematical and
  computational problems of modeling biological
  processes in the cell, removing experimental
  errors from genomic data, interpreting the data
  and providing theories about their biological
  relations
• Multidisciplinary field at the intersection of
  computer science, biology, discrete mathematics,
  statistics, optimization, chemistry, physics,

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5 Steps to Solving CB Problems

• Understand biological problem
• Represent biological data as mathematical objects
  (strings, sets, graphs, permutations,), map
  biological relations into mathematical relations,
  and formulate the biological question as
  optimization or feasibility problem
• Study computational complexity Polynomial?
  NP-hard?
• Develop efficient algorithms
• If in P, find fast and memory efficient exact
  algorithms
• If NP-hard, find practical exact algorithms
  and/or algorithms with provable approximation
  guarantees
• Validate algorithms on biological data

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Outline

• Shortest Superstring
• Sequencing by Hybridization
• PCR Primer Selection

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Shotgun Sequencing

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Shortest Superstring

• Given set of strings s1, s2, , sn
• Find shortest string s containing each si as a
  substring
• Example
• Set of strings 000, 001, 010, 011, 100, 101,
  110, 111
• Superstring 0001110100
• NP-Hard MaierStorer77

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Greedy Merging Algorithm

• S s1,s2,,sn
• While S gt 1 do
• Find s,t in S with longest overlap
• S ( S \ s,t ) U s overlapped with t to
  maximum extent
• Output final string

• Approximation factor no better than 2
• s1 abk, s2 bkc, s3 bk1
• Greedy output abkcbk1 length 2k3
• Optimum abk1c length k3
• Open problem prove that greedy superstring is
  always at most twice longer than optimum

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Overlap Prefix of 2 strings

• Overlap of s and t longest suffix of s that is a
  prefix of t
• Prefix of s and t s after removing overlap(s,t)
• s a1 a2 a3 as-k1as
• t b1 bk bt
• prefix(s,t)
• overlap(s,t)

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Lower Bound on OPT
OPT prefix(s1,s2) prefix(sn-1,sn)
prefix(sn,s1) overlap(sn,s1) cost
of tour 1?2??n in the prefix graph
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The Cycle Cover Algorithm

• Computing TSP in prefix graph is NP-hard
• Key idea lowerbound OPT using min-weight cycle
  cover
• For every cycle c (i1?i2??il?i1), ?(c)
  prefix(si1,si2) prefix(sil,si1) si1 is a
  superstring of si1, , sil
• Cycle cover algorithm

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The Cycle Cover Algorithm

• Theorem Blum,Jiang,Li,Tromp,Yannakakis94 Cycle
  cover algorithm gives factor 4 approximation.
• Length of output is
• where ri is a representative string from
  cycle ci
• wt(C) ? OPT
• - If ri no longer than wt(ci) ? output within
  factor 2 of optimum!
• ri can be much longer than wt(ci) (periodic
  strings!)
• it can be shown that ? ri ? OPT 2 wt(C) ?
  factor 4

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Improved Algorithm
Theorem Blum,Jiang,Li,Tromp,Yannakakis 94 The
improved algorithm gives factor 3
approximation. Proof using that the greedy
algorithm gives at least ½ of the optimum
compression. Current best approximation factor is
2.596 Breslauer,Jiang,Jiang97
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Sequencing by Hybridization

• Exploits parallel hybridization in DNA arrays
• All 4k probes of a certain length k (k8 to 10)
  are synthesized on the array
• Target DNA hybridizes at locations containing
  probes complementary to its k-substrings
• Sequencing by Hybridization (SBH) Problem
  Reconstruct target DNA given its k-length
  substrings (spectrum)

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Mathematical Formulation of SBH

• SBH is a special case of the shortest
  superstring solution corresponds to a
  Hamiltonian path (NP-hard to find) in the prefix
  length 1 graph
• Pevzner 89 SBH is equivalent to finding an
  Eulerian path (easy to find in linear time) in
  the following graph
• Vertices are all (k-1)-tuples
• Directed edge between two (k-1)-tuples u and v
  iff there is a k-length string in the spectrum
  whose first k symbols match u and last k symbols
  match v
• Choose the right mathematical abstraction!

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Polymerase Chain Reaction

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Primer Selection Problem
r i
3'
5'
Reverse primer
? Lx
? Lx
Forward primer
3'
5'
f i
i-th amplification locus

• Given
• Pairs of forward/reverse sequences for the n
  amplification loci
• Primer length k and amplification upperbound L
• Find
• Minimum set of primers S of length k such that,
  for each amplification locus, there are two
  primers in S hybridizing to the forward and
  reverse sequences within a distance of L of each
  other

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Previous Work

• Pearson et al. 96 Logarithmic approximation
  factor using greedy set cover algorithm for a
  formulation that does not distinguish between
  forward and reverse primers
• Similar formulations used by LinhartShamir02,
  Souvenir et al.03
• To enforce bound of L on amplification length
  must truncate forward and reverse sequences to
  length L/2
• FernandesSkiena02 model primer selection as
  a minimum multicolored subgraph problem
• Vertices are candidate primers
• Add edge colored by color i between primers u
  and v if they hybridize to i-th forward and
  reverse sequences within a distance of L
• Find minimum size set of vertices inducing edges
  of all colors
• No non-trivial approximation factor proposed

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Improved Approximations

• Konwar,M,Russell,Shvartsman 04
• Logarithmic approximation factor using
  potential function greedy for the bounded
  amplification length primer selection problem
• O(Lln n) approximation factor based on
  randomized rounding for the minimum multicolored
  subgraph problem of FernandesSkiena02

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Improved Approximations

• Konwar,M,Russell,Shvartsman 04
• Logarithmic approximation factor using
  potential function greedy for the bounded
  amplification length primer selection problem
• O(Lln n) approximation factor based on
  randomized rounding for the minimum multicolored
  subgraph problem of FernandesSkiena02

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Key Lemma
If r and r are representative strings from
cycles c and c, then
If overlap(r,r) ? wt(c) wt(c), then ??
(?)? ? ?? covers strings in both c and c ?
cycle cover is not minimal
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Proof of Factor 4

• Length of output
• Numbering ris in order of lefmost occurrence in
  OPT and using Lemma
• ? ? ri ? OPT ? overlap(ri,ri1) ? OPT
  2 wt(C)
• wt(C) ? OPT
• ? Length of output ? 4 x OPT

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Improved Algorithm Analysis

• Observation 1 The greedy algorithm is known to
  achieve at least ½ of the optimum compression,
  i.e.,
  ? ?(ci) - ? ? ½ (
  ? ?(ci) - OPT?)
  where OPT? is the shortest superstring of
  ?(ci), i1,,k
• ? ? - OPT? ? ½ (? ?(ci) - OPT?)
• Observation 2 By numbering ?(ci)s in order of
  lefmost occurrence in OPT? and using again the
  key Lemma
• ?(ci) - OPT? ? overlap(?(ci), ?(ci1)) ?
  2 wt(C)
• ? ? - OPT? ? wt(C)
• Observation 3 OPT? ? OPT wt(C)
• ? ? ? 3 OPT