1. Protein Folding in the 2D HP Model Alexandros Skaliotis – King’s College London • Joint work with: • Andreas Albrecht (University of Hertfordshire) • Kathleen Steinhöfel (King’s College London)

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

3. 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.

4. 2. Protein Folding • Protein Folding is the process by which a sequence of amino acids conforms to a three-dimensional shape. • Anfinsen’s 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.

5. 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 ξ < 0: Influence ratio of the new HH contacts (usually ξ = -1) • Objective Function = HHc * ξ = -HHc

6. 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.

7. 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

8. 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.

9. 6. Set of Moves 1 1 1 L 2 L 2 C 3 3 L 3 L 3 L 4 4 5 5 4 4 5 6 6 6

10. 7. Logarithmic Cooling Schedule • Following Hajek’s 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. • Cooling Function:

11. 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

12. 9.1 Experiment • Estimate experimentally. Processor: 2.2 GHz AMD Athlon

13. 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 .