Protein Folding in the 2D HP Model

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Protein Folding in the 2D HP Model

• Alexandros Skaliotis Kings College London

• Joint work with
• Andreas Albrecht (University of Hertfordshire)
• Kathleen Steinhöfel (Kings College London)

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Overview

• Proteins
• Protein Folding
• 2D HP Model
• Simple Example
• Local Search for Protein Folding
• Set of Moves
• Logarithmic Cooling Schedule
• Selected Benchmarks
• Experiment

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1. Proteins

• A protein is a sequence of amino acids encoded by
  a gene in a genome.
• There are 20 different amino acids.
• The length of the sequence can range from about
  20 to 3500.
• The function of a protein is determined by its
  three-dimensional structure.
• Predicting this structure is quite daunting and
  very expensive.

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2. Protein Folding

• Protein Folding is the process by which a
  sequence of amino acids conforms to a
  three-dimensional shape.
• Anfinsens hypothesis suggests that proteins fold
  to a minimum energy state.
• So, our goal is to find a conformation with
  minimum energy.
• We want to investigate algorithmic aspects of
  simulating the folding process.
• We need to simplify it.

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3.1 2D HP Model Dill et al. 1985

• Classify each amino acid as hydrophobic (H) or
  hydrophilic (P).
• Confine consecutive amino acids to adjacent nodes
  in a lattice (Treat search space as a grid).
• Flatten the search on a 2D lattice.
• Function HHc Number of new HH contacts
• Parameter ? lt 0 Influence ratio of the new HH
  contacts (usually ? -1)
• Objective Function HHc ? -HHc

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3.2 2D HP Model Dill et al. 1985

• Protein Folding in the 2D HP Model is NP-Hard for
  a variety of lattice structures
  Paterson/Przytycka 1996 Hart/Istrail 1997
  Berger/Leighton 1998 Atkins/Hart 1999.
• Constant factor approximations in linear time but
  not helpful for predictions of real protein
  sequences Hart/Istrail 1997.
• Exact methods work only for sequences up to
  double digits length.

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4. Simple Example

• Normally the energy is a positive number
• But we have a minimisation problem, so we talk
  about negative energies

H RED P PINK
Energy 0
Energy -3
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5. Local Search for Protein Folding

• A wide range of heuristics have been applied to
  find optimal HP structures, especially
  evolutionary algorithms.
• Lesh et. Al (2003) and Blazewicz et al. (2005)
  applied tabu search to the problem.
• We apply Logarithmic Simulated Annealing.
• To move in the search space we employ a complete
  and reversible set of moves proposed by Lesh et
  al. in 2003 and Blazewicz et al. in 2005.

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6. Set of Moves
1
1
1
L
L
2
2
C
3
3
L
L
3
3
L
4
5
4
5
4
4
5
6
6
6
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7. Logarithmic Cooling Schedule

• Cooling Function

• Following Hajeks theorem (1988), we are
  guaranteed to find the optimal solution after an
  infinite number of steps if and only if
  .
• is the maximum value of the minimal escape
  heights from local minima.
• Albrecht et al. show that after
  transitions, the probability to be in a minimum
  energy conformation is at least , where
  n is the maximum size of the neighbourhood of
  sequences.

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8. Selected Benchmarks

• S36 3P 2H 2P 2H 5P 7H 2P 2H 4P 2H 2P 1H 2P
• S60 2P 3H 1P 8H 3P 10H 1P 1H 3P 12H 4P 6H 1P
  2H 1P 1H 1P
• S64 12H 1P 1H 1P 1H 2P 2H 2P 2H 2P 1H 2P 2H 2P
  2H 2P 1H 2P 2H 2P 2H 2P 1H 1P 1H 1P 12H

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9.1 Experiment

• Estimate experimentally.

Processor 2.2 GHz AMD Athlon
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9.2 Experiment

• We found that is a good estimated
  upper bound for .
• We checked this against S85 and got the best
  known results in 10 / 10 runs.
• Of course we need more benchmarks.
• But this can be a good starting point in trying
  to develop a formal proof for the value of .