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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) Overview 1. 2. 3. 4. 5. 6. 7. 8. 9. Proteins Protein Folding 2D HP Model Simple Example Local Search for Protein Folding Set of Moves Logarithmic Cooling Schedule Selected Benchmarks Experiment 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 threedimensional structure. • Predicting this structure is quite daunting and very expensive. 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. 3.1 2D HP Model [Dill et al. 1985] 1. Classify each amino acid as hydrophobic (H) or hydrophilic (P). 2. Confine consecutive amino acids to adjacent nodes in a lattice (Treat search space as a grid). 3. 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 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. 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 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. 6. Set of Moves 1 L 2 C 3 L 4 1 2 L L 5 6 3 L 4 5 6 7. Logarithmic Cooling Schedule • Cooling Function: g c(k ) , k 0, 1,... ln( k 2) • Following Hajek’s theorem (1988), we are guaranteed to find the optimal solution after an infinite number of steps if and only if g . • is the maximum value of the minimal escape heights from local minima. O( g ) • Albrecht et al. show that after (n / ) transitions, the probability to be in a minimum energy conformation is at least 1 , where n is the maximum size of the neighbourhood of sequences. 8. Selected Benchmarks • • S36: S60: • S64: 3P 2H 2P 2H 5P 7H 2P 2H 4P 2H 2P 1H 2P 2P 3H 1P 8H 3P 10H 1P 1H 3P 12H 4P 6H 1P 2H 1P 1H 1P 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 9.1 Experiment • Estimate experimentally. 20 runs 20 runs 20 runs Sequence g n g n /2 g n /4 Time Frame S36 Optimal Optimal Optimal 10 min S60 Optimal Optimal No 30 min S64 Optimal Optimal No 90 min Processor: 2.2 GHz AMD Athlon 9.2 Experiment • We found that n / 2 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 .