Algorithms for Generalized Comparison of Minisatellites

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Algorithms for Generalized Comparison of
Minisatellites

• Behshad Behzadi Jean-Marc Steyaert
• LIX, Ecole Polytechnique
• France

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Outline

• Biology
• Evolutionary Model
• Problem description
• Previous works
• Algorithms
• Results

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Biology

• Minisatellites consist of tandem arrays of short
  repeat units found in genome of most higher
  eukaryotes.
• High degree of polymorphism at minisatellites has
  applications from forensic studies to
  investigation of origin of modern humans.

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Biology

• These repeats are called variants.
• MVR-PCR is designed to find the variants.
• As an example, MSY1 is the minisatellite
• on the human Y-chromosomes. There are five
  different repeats (variants) in MSY1.

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Different Repeat Types (Variants) of MSY1
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Graphical representations of Minisatellite Maps
of 13 males
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Summary

• Biologists are able to compute the minisatellite
  maps A sequence in which each of the repeats is
  replaced by its symbol.
• Study of evolution of minisatellites is an
  important problem, in human genetics studies.

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Computer Science Model

• Each variant is a symbol of an alphabet.
• A minisatellite is a string on this alphabet.
• We need to compare these strings.

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Evolutionary Operations

• Insertion
• Deletion
• Mutation
• Amplification (p-plication)
• Contraction (p-contraction)

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Examples of operations(1)

• Insertion of d
• abbc gt
  abbdc
• Deletion of c
• abbcb gt
  abbb
• Mutation of c into d
• caab gt daab
  

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Examples of operations(2)

• 4-plication of c
• abcb gt abccccb
• 2-contraction of b
• abbc gt abc
• No subword replication or contraction

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Cost Functions

• I(x) insertion of letter x
• D(x) deletion of letter x
• M(x,y) mutation of x to y
• Ap(x) p-plication of letter x
• Cp(x) p-contraction of letter x

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Hypotheses

• All the costs are positive.
• The cost of duplications (and contractions) is
  less than all other operations.
• Distance M(x,x)0
• Triangle Inequality holds
• M(x,y)M(y,z) M(x,z)

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Transformation of s into t

• Applying a sequence of operations on s
  transforming it into t.
• An example xyy gtgt xbcbxzc
• xyy gt xy gt xxy gt xxz gt xbxz gt xbbxz gt
  xbbbxz gt xbcbxz gt xbcbxzc
• The cost of a transformation is the sum of costs
  of its operations.

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Transformation distance between s and t

• TD Minimum cost for a possible transformation
  of s into t.
• The transformation which gives this minimum is
  called optimal transformation.

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Previous Works

• Jobling al.
• Bérard Rivals (RECOMB02)
• B.B. J.M.S. (CPM2003, WABI2004) this work

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Optimal Transformation between s and t

• For any transformation of s into t there are 2
  different types for the symbols of s.
• Generative vs Vanishing letters of s
• create a substring in t (generation) or
• disappear (reduction)

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Basic lemmas

• The optimal generation of a non-empty string s
  from a symbol x can be achieved by a
  non-decreasing generation.
• In an optimal transformation of a string s to a
  string t any contraction operation can be done
  before any generation.

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The schema of the proof

• Sequence u is eliminated sometime during the
  process
• The right-hand side transformation is equivalent
  and less expensive w.r.t. evolution.

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Optimal Transformation

• generative and vanishing symbols can be
  transformed in two distinct optimized phases.

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The Algorithm

• Preprocessing ( Substring generation costs) by
  Dynamic programming
• Main part (Transformation distance) by Dynamic
  programming

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Substring generation costs

• Gi, j, x minimum generation cost of ti..j
  from symbol x among all generations which do not
  start by a mutation.
• Ti,j,x is the minimum generation cost of
  substring ti..j from symbol x.
• mci,j,p,x is the minimum generation cost for
  generating ti..j from symbol x among all
  possible generations starting with a
  p-plication.

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mci, j, p, x
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Substring generation costs
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Substring Reduction cost

• Si,j is the minimum cost of reduction of the
  substring si..j into si.
• Si,j is determined in the same way.

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Complexity

• The time complexity is
• The space complexity is
• The maximum possible p for a p-plication is noted
  by

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Transformation Distance

• TDi,j is the transformation distance between
  s1..i and t1..j.

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Complexity

• The main algorithm complexity is O(n³) in time
  and O(n²) in space.
• The total time complexity is
• The total space complexity is

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Further improvements

• Improving the complexity using the
• Run Length Encoded string representation.
• The RLE of aaaabbbbcccabbbbcc
  is a4b4c3a1b4c2
  also written a4b4c3ab4c2
• The lengths of the encoded strings with original
  lengths m and n are denoted by m' and n'.

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Generation of Runs

• There exists an optimal generation of a non-empty
  string t from a single symbol x in which for
  every run of size k gt 1 in t the k-1 right
  symbols of the run are generated by duplications
  of the leftmost symbol of the run.

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New configurations in the transformation
Generations could split
runs into several parts... Similarly for
reductions... See on examples different
configurations
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PreProcessing Generation Costs

• Compute the generation cost of all substrings of
  the target string t from any symbol x of the
  alphabet.
• Fill a table Gt x,i,j by recurrence.

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Core Algorithm

• The Transformation Distances TD between s1..i
  and t1..j are computed by recurrence according
  to lemmas derived to the situations
• Generalized dynamic programming is used again
• Complexity O(n'3m'3mn'2nm'2mn)

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BS algorithms vs. BR algorithms

• Complexity improvement O(n4) to O(n3) and more
  with RLE (O(n2) experimentally)
• Generalization 1 amplifications and contractions
  of order gt 2
• Generalization 2 symbol-dependent cost functions
  
• The triangle hypotheses on cost functions are not
  restrictive and can be released by some
  preprocessing.

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Dataset

• Provided by Prof. M. Jobling
• Minisatellite maps of 690 Y chromosomes from
  worldwide population.
• The length of the sequences is between 48 and
  118.
• Distances were computed for 690x690 pairs

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Conclusion

• More efficient algorithm to compute faster the
  distances and thus the phylogenetic trees.
• A more general framework which can be used for
  modelling more complicated biological evolutions.
  

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Thank you