Geometric Crossover for Biological Sequences

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EuroGP 2006
Geometric Crossover for Biological Sequences
Alberto Moraglio, Riccardo Poli Rolv
Seehuus
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Contents

• Geometric Crossover
• Geometric Crossover for Sequences
• Is Biological Recombination Geometric?

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I. Geometric Crossover
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Geometric Crossover

• Representation-independent generalization of
  traditional crossover
• Informally all offspring are between parents
• Search space all offspring are on shortest paths
  connecting parents

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Geometric Crossover Distance

• Search Space is a Metric Space d(A,B) length of
  shortest paths between A and B
• Metric space all offspring C are in the segment
  between parents
• C in A,Bd ?? d(A,C)d(C,B)d(A,B)

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Example1 Traditional Crossover

• Traditional Crossover is Geometric Crossover
  under Hamming Distance

Parent1 011101 Parent2 010111 Child
011111
HD(P1,C)HD(C,P2)HD(P1,P2) 1 1
2
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Example2 Blending Crossover

• Blending Crossover for real vectors is geometric
  under Euclidean Distance

ED(P1,C)ED(C,P2)ED(P1,P2)
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Many Recombinations are Geometric

• Traditional Crossover for multary strings
• Box and Discrete recombinations for real vectors
• PMX, Cycle and Order Crossovers for permutations
• Homologous Crossover for GP trees
• Ask me for more examples over a coffee!

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Being geometric crossover is important because.

• We know how the search space is going to be
  searched by geometric crossover for any
  representation convex search
• We know a rule-of-thumb on what type of
  landscapes geometric crossover will perform well
  smooth landscape
• This is just a beginning of general theory, in
  the future we will know more!

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II. Geometric Crossover for Sequences
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Sequences Edit Distance

• Sequence variable-length string of character
  from an alphabet A
• Edit distance minimum number of edit operations
  insertion, deletion, substitution to
  transform one sequence into the other
• A a,c,t,g, seq1 agcacaca, seq2 acacacta
• Seq1agcacaca ? acacacta ? acacactaSeq2
• ED(Seq1,Seq2)2 (g deleted, t inserted)

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Sequence Alignment (on contents)

• Alignment put spaces (-) in both sequences such
  as they become of the same length
• Seq1 agcacac-a
• Seq2 a-cacacta
• Alignment Score number of mismatches 2
• Optimal alignment minimal score alignment (Best
  Inexact Alignment on Contents)
• The score of the optimal alignment of two
  sequences equals their edit distance
  ED(Seq1,Seq2)Score(A)2

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Homologous Crossover

• Align optimally two parent sequences
• Generate randomly a crossover mask as long as the
  alignment
• Recombine as traditional crossover
• Remove dashes from offspring

Mask 111111000 Seq1 agcacac-a Seq2
a-cacacta SeqC a-cacac-a SeqC acacaca
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Theorem Geometricity of HC

• Homologous Crossover is geometric crossover under
  edit distance
• Seq1agcacaca ? SeqCacacaca ?acacactaSeq2
• ED(Seq1,SeqC)ED(SeqC,Seq2)ED(Seq1,Seq2)
• 1 1
  2

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More theory on HC in the paper

• Extension to weighted edit distances Extension to
  block ins/del edit distances
• Peculiarity of metric segments in edit distance
  spaces
• Bounds on offspring size due to parents size

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III. Is Biological Recombination Geometric?
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Recombination at a molecular level

• DNA strands align on the contents, no
  positionally
• DNA are flexible, can be stretched or folded to
  align better to each others
• DNA strands do not need to be aligned at the
  extremities
• Some pair matching are preferred to others
• DNA strands can form loops
• Crossover points happen to be where DNA strands
  align better
• Not all details worked out yet!

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Homologous Crossover as a Model of Biological
Recombination
Many possible variants of edit distance that fit
many real requirements of biological
recombination
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Minimum Free Energy Edit Distance

• DNA strands align optimally according to edit
  distance because
• (i) The alignment of two DNA strands
  (macromolecules) obeys chemistry it is the state
  at minimum free energy
• (ii) The weights of the edit moves can be
  interpreted as repulsion forces at a single basis
  level
• (iii) The best alignment on edit distance is the
  best trade-off for which the global effect of
  repulsion forces is minimized the minimum free
  energy alignment

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Is Biological Recombination Geometric? Yes?!
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So what?
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Bridging Natural and Artificial Evolution

• Bridging Natural and Artificial Evolution
• into a common theoretical framework
• Change in perspective this allows to study real
  biological evolution as a computational process
• In the paper we use geometric arguments to claim
  that biological evolution does efficient
  adaptation!

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Summary

• Geometric crossover
• Geometric crossover offspring between parents
• Many recombinations are geometric
• Some general theory for geometric crossover
• Homologous crossover
• Homologous crossover for sequences alignment on
  contents before recombination
• Homologous crossover is geometric under edit
  distance
• Biological Recombination
• Homologous crossover models biological
  recombination at DNA level, so it is geometric
• Geometric theory applies to biological
  recombination, bridging biological artificial
  evolution

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Questions?