Finding Consensus in a Family of DNA Sequences

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Finding Consensus in a Family of DNA Sequences

• Joshua W. Gilkerson

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Agenda

• Consensus Sequences
• Combinatorial Optimization
• Simulated Annealing
• Genetic Algorithms
• Application to Finding a Consensus Sequence

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Consensus Sequence

• A consensus sequence is one that captures the
  important features of a family of sequences.
• Consensus sequences have many important
  applications in bioinformatics.
• Used to define the real sequence of a feature
  that may appear several times in a genome,
  slightly modified.
• Genes
• Repeats

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An Example

• Form a consensus sequence from the following
  strings
• THE LONGER SEQUENCE
• A SEQUENCE
• THIS TISAS
• S RIS ANEQ

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An Example

• THE LONGER SEQUENCE
• A SEQUENCE
• THIS TISAS
• S RIS ANEQ

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Definition Point Mutation

• A modification to a string at a single location,
  changing a single character.
• May be one of
• Insert
• Delete
• Substitute

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Definition Levenshtein Distance

• The minimum number of point mutations required to
  transform one string into another.
• Also known as the edit distance.

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Definition Consensus Sequence

• The most widely used definition of a consensus
  string is that it minimizes the sum of the edit
  distances to the strings in the family.
• Also known as the Steiner Consensus String
• Very similar to a mean, which minimizes the sum
  of the distances to a family of numbers.

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Why Not An Exact Algorithm

• The determination of a consensus sequence has
  been shown to be NP-hard (Gusfield).
• Instances from bioinformatics are large, making
  them intractable.
• Therefore, an approximation technique is needed.
• Simulated Annealing and Genetic Algorithms have
  been applied successfully to other NP-hard
  problems.

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Combinatorial Optimization

• The process of finding minimum (or maximum)
  values for a function whose domain space contains
  many independent variables.
• Finding a consensus sequence is a type of
  combinatorial optimization, where the value to be
  minimized is the sum of the edit distances.

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Simulated Annealing

• is a process developed for other combinatorial
  optimization problems (Kirkpatrick), and has
  shown promise for finding consensus sequences
  (Keith).
• approximates the process of annealing used to
  produce very dense, very stable masses.
• Annealing involves heating a mass and the
  allowing it to cool very slowly. The product is
  an object whose molecules are arranged to have
  the lowest possible energy.

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Simulated Annealing

• Start with an initial approximation and an
  initial temperature.
• Iteratively, choose a set of adjacent
  approximations.
• Randomly choose the next approximation from this
  set plus the current approximation, weighted
  using the value begin optimized.
• Slowly lower the temperature.

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Simulated Annealing

• Sequences are weighted proportionally to the
  expression
• kB is the Boltzmann constant
• T is the current temperature
• ?E is the change in the target variable

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Simulated Annealing

• As the temperature becomes lower, the odds of
  choosing a higher energy state also become lower.
• This allows the approximation to move freely at
  first, but even at high temperatures, if a very
  low energy position is found, it is hard to leave.

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An Example

• This is a very simplified example where the
  domain is a single variable.
• The value to be optimized is plotted to the left.

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An Example

• If moves to higher energy are not allowed,
  starting anywhere between 2 and 4 would yield the
  final answer of 3, which does not have minimum
  energy.

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An Example

• However, if moves to higher energy states are
  allowed, it is at least possible to find the
  optimal answer of 5, no matter what the initial
  approximation is.

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Application to Consensus Sequences

• While many methods have been developed to find
  consensus sequences, most are expensive and dont
  scale well to larger numbers of sequences.
• Most prior methods involve doing multiple
  sequence alignments, and using these alignments
  to form a consensus, however using simulated
  annealing finds a solution directly.
• The consensus sequence may then be used to
  quickly generate a multiple alignment if needed.

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Application to Consensus Sequences

• Sum of edit distances used as optimization value.
• Sets of optimizations are constructed by
  iteratively considering each position in the
  current approximation and making all possible
  point mutations at that position.

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Analysis

• In the algorithm as outlined here and presented
  fully by Keith et al, the runtime
• scales linearly with the number of sequences
• scales quadratically with the length of the
  sequences
• Empirical results have verified these theoretical
  predictions.

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Genetic Algorithm

• Is another process developed for solving
  combinatorial optimization problems.
• Approximates the process of evolution by natural
  selection as seen in nature

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Genetic Algorithm

• Start with an initial population (either random
  or some domain specific approximation).
• Generate offspring from the initial population.
• Remove the less fit offspring.
• Fitness if measured by the value being optimized.

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Genetic Algorithm

• Generating offspring
• Can be done sexually or asexually
• Sexual reproduction takes part of the candidate
  solution from each of two parents already in the
  population.
• Parts of the candidate may then be mutated.

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Genetic Algorithm

• Because of this process of producing offspring,
  problems whose solution space can be represented
  as a vector (or string) are most easily
  addressed.
• Sexual reproduction allows for some large jumps
  within the solution space, but limited by the
  current population.

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An Example

• Same example as used earlier.

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Application to Consensus Sequences

• Initial population is the set of strings under
  consideration
• Random chance of sexual or asexual reproduction.
• If sexual, a prefix is selected from one sequence
  and a suffix from another. The lengths are
  selected randomly over a poisson distribution to
  favor offspring of approximately the same length
  as the parents.
• Each position of the sequence is mutated with
  some probability.

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Application to Consensus Sequences

• A number of offspring are produced, then some
  number of the best are kept.
• The process is repeated some number of times.

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Analysis

• In the algorithm as outlined, the runtime
• Scales linearly with the number of sequences
• Scales quadratically with the length of the
  sequences
• Empirical results have suggested some interesting
  improvements for setting the population sizes and
  the number of generations.
• Speculatively, I expect a final versions runtime
  to
• Scale quadratically with the number of sequences.
• Scale cubucally with their lengths.

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Summary

• Simulated annealing and genetic algorithms have
  been used successfully in many other
  combinatorial optimization problems including the
  traveling salesman problem.
• Likewise, early results are promising for finding
  consensus sequences.
• No guarantees of accuracy.
• However, it appears that it might be possible to
  give some approximation of how accurate a given
  solution is.
• Very fast.

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References

• Day, W. H., and F. R. McMorris. The computation
  of consensus patterns in DNA sequence.
  Mathematical and Computer Modelling.
  17(1993)49-52.
• Keith, Jonathan M., Peter Adams, Darryn Bryant,
  Dirk P. Kroese, Keith R. Mitchelson, Duncan A.E.
  Cochran, and Gita H. Lala. A Simulated Annealing
  Algorithm for Finding Consensus Sequences.
  Bioinformatics. 18(2002)1495-9.
• Kirkpatrick, S., C. D. Gelatt, Jr., M. P. Vecchi.
  Optimization by Simulated Annealing. Science.
  220(1983)671-80.