Nonbreaking Similarity of Genomes with Gene Repetitions

1
Non-breaking Similarity of Genomes with Gene
Repetitions

• Binhai Zhu
• Computer Science Department, Montana State
  University
• Joint work with Zhixiang Chen, Bin Fu, Jinhui
  Xu, Boting Yang and Zhiyu Zhao

2
Background

• Computing genomic distance between genomes is
  important in evolutionary molecular biology, the
  problem was first studied by Sturtevant and
  Dobzhansky in 1936.
• A lot of research has been done on computing
  genomic distances since 1990, assuming that each
  gene appears in a genome once, e.g., the famous
  result by Hannenhalli and Pevzner on sorting
  signed permutations by reversals.

3
Background (cond.)

• On the other hand, gene repetition is very common
  in genomes. So computing genomic distances with
  gene repetition is a more realistic problem.
• This is a typical optimization problem, it makes
  sense to study the approximability of the
  problem.

4
Definitions

• Given n gene families (alphabets) F, a genome G
  is a sequence of elements of F such that each
  element has a (/-) sign.
• Example. Fa,b,c,d,
• G-bd-cab-d-c
• We will focus on unsigned sequences in this work.
• A genome G is said to be exemplar if every gene
  appears exactly once in G.

5
Definitions (cond.)

• Given exemplar genomes G and H, over the same
  set of gene families, if gene ab is a substring
  in G but not in H, then ab constitutes a
  breakpoint in G.
• Example, Gabcdefg
• Hefgdcab
• there are 3 breakpoints in G (and
  symmetrically in H).
• The number of breakpoints between G and H is
  called the breakpoint distance between G and H.

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Exemplar Breakpoint Distance Problem

• Given two genomes G and H over n gene families,
  compute two exemplar genomes G and H such that
  the breakpoint distance between G and H is
  minimized.
• We call this the exemplar breakpoint distance
  problem (between G and H). Denote this distance
  by eb(G,H)b(G,H).

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Approximation Algorithms

• Given a minimization (maximization) problem ?,
  let the optimal solution of ? be OPT, an
  approximation algorithm A provides a performance
  guarantee of a for ? if for every instance of ?
  the solution value returned by A is at most ? x
  OPT (at least OPT/?).
• Usually we say that A is a factor-? approximation
  for ?.

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Prior Results (1)

• We showed that the exemplar breakpoint distance
  problem does not admit any approximation, unless
  PNP (or, deciding whether eb(GH)0 is
  NP-complete) Chen, Fu and Zhu2006.
• This result holds for any genomic distance d( )
  satisfying GH implies d(G,H)0.
• Based on the above result, even under a weaker
  model of approximation, we showed that the
  exemplar conserved interval distance problem does
  not admit any WEAK approximation of a superlinear
  factor Chen, Fowler, Fu and Zhu, 2007.

9
Prior Results (2)

• On the other hand, for the exemplar breakpoint
  distance problem, Sankoff has used
  branch-and-bound Sankoff, 1999 and Nguyen, Tay
  and Zhang 2005 have used divide-and-conquer on
  practical datasets to obtain good empirical
  results.
• As a related, but slightly different effort,
  Chauve, et al. 2006 studied the exemplar
  genomic similarity problems which does not
  satisfy GH implies d(G,H)0, e.g., the exemplar
  common interval measure problem.

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Background for this work

• We try to look at the complement of the
  breakpoint distance under the gene duplication
  model.
• As the problem is still hard to approximate, we
  follow Nguyen, et al. by considering genomes
  satisfying some practical conditions.

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Definitions

• Given exemplar genomes G and H drawn from the
  same alphabet, ab is a non-breaking point, if ab
  appears in both G and H.
• Example. G abcdefg
• H fegcdab
• We have two non-breaking points in G and H,
  which is called the non-breaking similarity of G
  and H, denoted as nbs(G,H).
• Note that when GHn, if GH,
  nbs(G,H)n-1.
• Given genomes G and H drawn from the same
  alphabet, possibly with gene repetitions, the
  exemplar non-breaking similarity problem is to
  delete redundant genes to obtain exemplar genomes
  G and H such that nbs(G,H) is maximized. The
  corresponding measure is also denoted as
  enbs(G,H).

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Example

• G abcadcefg
• H cfegcdabf
• We have 4 possible exemplar genomes for G
  abcdefg, abdcefg, bcadefg, badcefg.
• We have 4 possible exemplar genomes for H
  cfegdab, cegdabf, fegcdab, egcdabf.
• enbs(G,H)nbs(abcdefg,fegcdab)2.

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Inapproximability Result

• Theorem 1. Given an exemplar genome G and
  another genome H such that the genes are all
  from the same alphabet with size n and each gene
  appears in H at most two times, the Exemplar
  Non-breaking Similarity Problem over G and H
  does not admit any approximation of factor n1-e,
  unless PNP.
• Proof Idea A linear reduction from Independent
  Set (IS).

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e2
v2
v1
N5 vertices, M5 edges NM is even
e4
e3
e1
e5
v4
v5
v3
Gv1v1v2v2v3v3v4v4v5v5x1e1x1x2e2x2x3e3x3x4
e4x4x5e5x5
HYNM-1YNM-3Y1YNMYNM-2Y2
x4x4x2x2v5e4e5v5v3e1v3v1e1e2v1x5x5x3x3x1x1
v4e3e5v4v2e2e3e4v2 YiviAivi, if i N
YNixixi, if iM
Hx4x4x2x2v5e5v5v3v3v1e1e2v1x5x5x3x3x1x1v4
v4v2e3e4v2 correspond to the optimal
independent set v3,v4
Input graph has an IS of size K iff enbs(G,H)K.
15
e2
v2
v1
N5 vertices, M5 edges NM is even
e4
e3
e1
e5
v4
v5
v3
Gv1v1v2v2v3v3v4v4v5v5x1e1x1x2e2x2x3e3x3x4
e4x4x5e5x5
HYNM-1YNM-3Y1YNMYNM-2Y2
x4x4x2x2v5e4e5v5v3e1v3v1e1e2v1x5x5x3x3x1x1
v4e3e5v4v2e2e3e4v2 YiviAivi, if i N
YNixixi, if iM
Hx4x4x2x2v5e5v5v3v3v1e1e2v1x5x5x3x3x1x1v4
v4v2e3e4v2 correspond to the optimal
independent set v3,v4
Input graph has an IS of size K iff enbs(G,H)K.
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Positive Results

• Our motivation was from Nguyen, Tay and Zhang
  2005, who observed that for certain bacteria
  genome pairs (Baphi-Wigg, Pmult-Hinft,
  Ecoli-Styphi, Xaxo-Xcamp and Ypes), repeated
  genes are usually pegged, e.g.,
• xyxaba

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Positive Results

• Definition
• occ(g,G) is the number of occurrence of g in
  G.
• span(g,G) is the maximum distance between two
  copies of g in G.
• totalocc(c,G)?gene g in G with span(g,G)c
  occ(g,G)

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Positive Results

• Definition
• occ(g,G) is the number of occurrence of g in
  G.
• span(g,G) is the maximum distance between two
  copies of g in G.
• totalocc(c,G)?gene g in G with span(g,G)c
  occ(g,G)
• Example. Gabcdaebd
• span(a,G)4, span(b,G)5, span(d,G)4,
• totalocc(4,G)6

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Positive Results

• Theorem 2. Let G and H be two genomes with
  ttotalocc(1,G) totalocc(c,H), for a constant
  c. Then enbs(G,H) can be computed in
  O(3t/3nc2e) time.

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Positive Results

• Theorem 2. Let G and H be two genomes with
  ttotalocc(1,G) totalocc(c,H), for a constant
  c. Then enbs(G,H) can be computed in
  O(3t/3nc2e) time.
• Idea 1 Given an exemplar genome G and another
  genome H satisfying span(g,H)c, for every g in
  H, we can use divide and conquer to compute
  enbs(G,H) in O(nc2e) time.
• Roughly speaking, HH1H2H3, H2c, then
  enumerate all solutions on H2 and recurse.
• T(n) 2c12T(n/2c) O(n) O(nc2e)

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Positive Results

• Theorem 2. Let G and H be two genomes with
  ttotalocc(1,G) totalocc(c,H), for a constant
  c. Then enbs(G,H) can be computed in
  O(3t/3nc2e) time.
• Idea 2 As t is considered as a constant, we
  enumerate all possibilities for deleting
  duplicated genes in G (to obtain G) and for
  deleting genes with span greater than c in H (to
  obtain H). By Lemma 6, there are at most
  4?3t/3 such combinations. Therefore, the total
  running time is 4?3t/3?O(nc2e)
  O(3t/3nc2e) time.

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Positive Results

• Theorem 3. Let G and H be two genomes with a
  total of t genes g satisfying shift(g,G,H) gtc,
  for some constant c. Then enbs(G,H) can be
  computed in O(3t/3n2c1e) time.
• Example. Gabcadef
• Hbcedefad
• shift(a,G,H) 6

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Conclusion

• We introduce non-breaking similarity, which is
  the complement of the famous breakpoint distance,
  for genome comparison.
• The general exemplar non-breaking similarity
  problem is hard to approximate.
• 3. For some special cases, we can obtain
  polynomial solutions.